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SPURIOUS PREDICTORS IN RANDOM COEFFICIENT MODELING
By

Michael Thomas Braun

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS
Psychology

2009

ABSTRCT
SPURIOUS PREDICTORS IN RANDOM COEFFICIENT MODELING
By
Michael Thomas Braun

A model fit to a longitudinal process results in a trajectory or set of trajectories.
When variability is present in the longitudinal growth process, researchers will typically
use time-varying covariates to predict the observed heterogeneity. Random coefficient
modeling such as Hierarchical Linear Modeling (HLM) is currently the dominant
approach to the analysis of longitudinal data in psychology because of its ability to
effectively deal with heterogeneous data while at the same time allowing researchers to
insert predictors into the model. The application of random coefficient models to
longitudinal data assumes that the psychological process under investigation results
solely from a deterministic trend. However, if a process, at least partially, results from a
stochastic trend, then, random coefficient regression results are spurious. Previous
research on simple regression models and Monte Carlo simulations are used to
demonstrate the spurious results across six commonly observed models. A data analytic
strategy is proposed to help researchers identify potential stochastic processes to avoid
making inaccurate statistical and scientific inferences. Finally, two statistical techniques
are briefly explained that can effectively handle stochastic data and help researchers

accurately model the longitudinal process under investigation.

ACKNOWLEDGEMENTS

I would like to extend my extreme gratitude to my advisor, Dr. Rick DeShon, for
all of the time, effort, and support that he has given me throughout this research process.
I am undoubtedly a better writer and researcher thanks to his guidance.

I would also like to thank my committee members, Neal Schmitt and Brent
Donnellan, for providing excellent guidance and insight on improving my current project.

Finally, thanks to all of my friends and family for all of the loving support they
have always provided me. A special thanks to my parents for their continued love,

encouragement, patience, and guidance.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. vi
LIST OF FIGURES .......................................................................................................... vii
INTRODUCTION .............................................................................................................. 1
Longitudinal Data Collection .................................................................................. 3
Longitudinal Data Analysis .................................................................................... 9
The Problem Presented ......................................................................................... 12
Random Walks and Stochastic Processes ............................................................. 13
Spurious Regression ............................................................................................. 18
RCM Conceptual and Mathematical Overview .................................................... 20
Spurious Regression in RCM ............................................................................... 25
Fixed Effects ............................................................................................. 26
Significance Tests on the Fixed Effects .................................................... 26
Variance Components for the Random Effects ........................................ 27
Significance Tests on the Variance Components for the Random
Effects ....................................................................................................... 28
Deviance Statistics .................................................................................... 30
Model 1: Unconditional Means Model ..................................................... 31
Model 2: Unconditional Growth Model ................................................... 33
Model 3: Unconditional Growth Model with X as a Fixed Effect ........... 35
Model 4: Unconditional Growth Model with X as a Fixed and Random
Effect ......................................................................................................... 36
Model 5: Unconditional Means Model with X as a Fixed Effect ............. 38
Model 6: Unconditional Means Model with X as a Fixed and Random
Effect ......................................................................................................... 40
METHOD ......................................................................................................................... 42
RESULTS ......................................................................................................................... 44
Model 1: Unconditional Means Model ................................................................. 44
Model 2: Unconditional Growth Model ............................................................... 45
Model 3: Unconditional Growth Model with X as a Fixed Effect ....................... 48
Model 4: Unconditional Growth Model with X as a Fixed and Random Effect.. 51
Model 5: Unconditional Means Model with X as a Fixed Effect ......................... 55
Model 6: Unconditional Means Model with X as a Fixed and Random Effect 57
General Findings .................................................................................................. 60
DISCUSSION ................................................................................................................... 62
Recommendations ................................................................................................ 62
Application to Latent Growth Models .................................................................. 65
Limitations and Future Directions ........................................................................ 68

Conclusion ............................................................................................................ 69

APPENDIX A: Standard Errors as Approximations of Standard Deviations of the
Sampling Distributions of the Fixed Effects ..................................................................... 71

REFERENCES ................................................................................................................. 7S

LIST OF TABLES

Table 1. Traditional Longitudinal Study Statistics ............................................................ 5
Table 2. Event/Experience Sampling Study Statistics ....................................................... 7
Table 3. Model 1 - Unconditional Means Model ............................................................. 45
Table 4. Model 2 - Unconditional Growth Model - Fixed Effect Statistics .................... 46
Table 5. Model 2 - Unconditional Growth Model - Random Effect Statistics ................ 47
Table 6. Model 3 - UGM with X as a Fixed Effect - Fixed Effect Statistics ................... 49
Table 7. Model 3 - UGM with X as a Fixed Effect - Random Effect Statistics .............. 50

Table 8. Model 4 - UGM with X as a Fixed and Random Effect - Fixed Effect
Statistics ............................................................................................................. 53

Table 9. Model 4 - UGM with X as a Fixed and Random Effect - Random Effect
Statistics ............................................................................................................. 54

Table 10. Model 5 - Unconditional Means Model with X as a Fixed Effect ................... 56

Table 11. Model 6 - UMM with X as a Fixed and Random Effect - Fixed Effect
Statistics ........................................................................................................... 58

Table 12. Model 6 - UMM with X as a Fixed and Random Effect - Random Effect
Statistics ........................................................................................................... 59

vi

LIST OF FIGURES

Figure 1. Random Walk ................................................................................................... 15
Figure 2. Deterministic and Stochastic Trends ................................................................ 16
Figure 3. Independent Random Walks ............................................................................. 19

vii

INTRODUCTION

Longitudinal data structures are increasingly common as researchers focus
attention on the dynamics of psychological processes. Along with the increase of
longitudinal data structures in psychology, the number Of time points being collected is
increasing (Walls & Schafer, 2006). When utilizing longitudinal data to study
psychological phenomena over time, two questions are frequently posed. First, how does
the criterion of interest change or grow over time and second, what variables can help
predict or explain the observed patterns in growth trajectories. When examining growth
curves to answer the first question, it is quite common to encounter a great deal of
heterogeneity across groups or individuals in the growth process (Collins & Sayer, 2001).
Researchers then typically attempt to answer the second question by searching for
predictors to explain the heterogeneity in growth. One common method to explain this
heterogeneity is to examine the relationship between the outcome of interest and a time-
varying covariate (predictor) over time. Random Coefficient Modeling (RCM), such as
multilevel models or Hierarchical Linear Models, can effectively deal with any clustering
(heterogeneity) that may exist, while at the same time allowing researchers to insert
predictors into the model to account for unexplained within- or between-group variance
(Gelman & Hill, 2006). As a result, RCM has become the most common analytical tool
for psychologists when analyzing longitudinal data.

A model fit to a longitudinal process results in a trajectory or set of trajectories.
Typically, in psychological research the observed trajectories are modeled as if they are

the result of a noisy, purely deterministic process. Although not widely recognized in

psychology, it is well known in other disciplines (e.g., economics, physics, and biology)
that the observed trends in these trajectories may be due, at least in part, to a stochastic,
or random, process. A problem can arise when using random coefficient modeling to
model growth data because the regression model and generalizations of that model, such
as RCM, require the assumption that all trends in the dependent variable result solely
from a deterministic process. When the regression model is used to analyze data resulting
at least partially from a stochastic process, a serious inflation Of Type I error rates, called
spurious regression, is often Observed. Spurious results can occur when regressing one
stochastic series on another, regressing a stochastic series on time, and when regressing a
stochastic series onto time and other stochastic predictors (Granger & Newbold, 1974;
Nelson & Kang, 1984). This issue is particularly problematic in psychological research
because frequently the focus of longitudinal studies is on growth processes, where
stochastic trends have been frequently discovered in other fields such as computer
science (Tang, J in, & Zhang, 2008), physics (Uhlenbeck & Omstein, 1930), genetics
(Wright, 1931), and economics (Nelson & Plosser, 1982). Since RCM is a generalization
of the regression model, it is likely to encounter the spurious results that are known to
occur when using regression to analyze stochastic processes. This is especially true when
time or Level 1, time-varying covariates are in the model as predictors (Nelson & Kang,
1984)

This proposal is organized in the following manner: I will demonstrate the
problem by outlining current trends in longitudinal data collection, discuss the issue of
heterogeneity in longitudinal data, and explain various methods for analyzing

longitudinal data concluding with a discussion of RCM, the most commonly used

analytical method. I will then discuss random walks where spurious regression was first
documented. In my discussion of random walks I will explain the concepts of
deterministic and stochastic processes, conceptually and mathematically explain what a
random walk is, and then outline multiple ways spurious regression is commonly
Observed with random walks. I will then conceptually and mathematically explain RCM
and its common applications. Next, I will apply the case of spurious regression to RCM
and discuss what can happen when researchers use RCM to evaluate the relationship
between stochastic series. Monte Carlo simulations will then be performed to
demonstrate the large inflation of Type I errors. Finally, I will discuss the implications of
not accounting for stochastic processes and will conclude by suggesting possible ways to

deal with this problem.

Longitudinal Data Collection

In recent years, many psychological methodologists have called for the use of
time in models, research designs, and theoretical frameworks (e.g., Ancona, Okhuysen, &
Perlow, 2001; George & Jones, 2000; McGrath & Rotchford, 1983; Mitchell & James,
2001). Along with the literature on methods and theory building, researchers in many
substantive areas in psychology have also called for the inclusion of time in theory and
models. Teams and multilevel theory are two of the most predominant areas in
organizational psychology where this has taken place (e.g., Kozlowski & Ilgen, 2006;
Kozlowski & Klein, 2000; Mohammed, Hamilton, & Lim, In Press).

Unfortunately, many challenges are encountered when trying to study groups or

individuals over an extended period of time. Traditional longitudinal studies utilized

parameters similar to those of most cross section designs, large sample sizes and very few
time points. Table l showcases some examples of these longitudinal designs from the last
fifieen years. These studies were randomly sampled from a large pool of journals using
the PsycINFO database. On average these studies had very large samples (IV = 1662) but
relatively few time points (7‘ = 6). The only exceptions in these traditional longitudinal
designs are large national studies such as the Minnesota Twin Study and the National
Longitudinal Survey of Youth. These two large scale national studies were chosen
because they are representative of ongoing national surveys used to address
psychological and other phenomena over time. Like most traditional longitudinal designs,
these large national studies focused more heavily on sample size (in the tens of
thousands), but they were also able to collect data over a relatively large number of time
points (7' = 24). With these types of parameters the focus of traditional longitudinal
designs is largely on number of participants rather than number of time points. This is
helpful for studying phenomena that are inherently between-person, however it still does
not provide a good method for studying phenomena that are within-person. Therefore, to
better understand intraindividual psychological phenomena over time, a different type of

longitudinal design was needed.

Table 1. Traditional Longitudinal Study Statistics

 

Traditional Longitudinal Designs

 

 

 

 

 

 

 

 

 

 

 

Author(s) Year N T Journal

Journal of Youth and
Baldwin & Hoffman 2002 762 l 1 Adolescence

International Journal of
Grimm 2007 7078 7 Behavioral Development

Journal of Studies on
Harford & Munthen 2001 2465 3 Alcohol
Hoffman et al. 2000 651 4 Substance Use and Misuse
Jang et al. 2004 320 4 The Gerontologist

Journal of Quantitative
Johnson et al. 1997 765 3 Criminology

Journal of Consulting and
Raudenbush & Chen 1993 1725 8 Criminal Psychology
Welte et al. 2005 625 3 American Journal of Drug

and Alcohol Abuse
Witkiewitz 2008 563 7 Psychology of Addictive

Behaviors

 

 

 

 

 

Note. N = Sample Size; T = series length.

Psychological phenomena naturally occur and develop within individuals. It is
therefore important to study and analyze these constructs at the individual level (Block,
1995). In fact, some psychological phenomena, for example self—efficacy, appear to
behave differently when studied at the within-person level compared to the between-
person level (Vancouver, Thompson, & Williams, 2001; Vancouver, More, & Yoder,
2008). These ideas have gained much support in recent years and there is now a push to
return the individual back into scientific psychology (Curran, Wirth, Nesselroade,
Rogosa, Thum, Tuerlinckx, & von Eye, 2004; Molenaar, 2004). Fortunately, along with
the push from psychological scholars for more longitudinal designs and research
questions focused on the individual came many technological advances to make the
process of collecting and analyzing longitudinal data easier for researchers. Computer

5

 

storage capacity has greatly increased over the last 20 years making it possible to
organize and keep data from many individuals over long periods of time in a relatively
cheap and easy way (Walls & Schafer, 2006). Statistical packages are now widely
available that have the most cutting edge analytical techniques making it easy and
convenient for researchers to run their analyses. Although the increased ability to store
and analyze longitudinal data was helpful, possibly the most influential technological
improvement on longitudinal data collection is the rise of cell phones and other wireless

computer devices.

This advancement in personal computers led to a rapid increase in the use of
event/experience sampling. Event sampling began in the form of diary studies and
evolved into a technique that utilizes some type of personal data device (e.g., palm pilot,
personal data assistant, cell phone, etc.) to randomly (at the researchers’ control) collect
information from participants multiple times a day for a given period of time. The data
collected are typically stored directly on the device and then downloaded to a computer at
the conclusion of the study. The rise of event sampling with personal computers makes
collecting longitudinal data much easier and more reliable for researchers and much more
convenient for participants. This allows researchers to focus on having fewer participants
with much longer series of data. Table 2 shows a sample of event sampling studies
published in APA journals in 2007 and 2008 gathered using PsyclNFO. As seen in the
table, the trend is to have relatively few participants (IV = 123) but have a large number of
data points for each person (7‘ = 47). Focusing on longer data streams for each person
allows researchers to get a clearer and more complete picture of the processes by which

psychological phenomena unfold over time within individuals. The push to study

psychological phenomena at the individual level and all of the benefits and insights
gained from this type of design has caused the number of event sampling studies to

greatly increase over the last few years (Walls and Schafer, 2006).

Table 2. Event/Experience Sampling Study Statistics

 

Ex erience/Event Sampling Studies

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Author(s) Year N T Journal
Journal of Research on
Bohnert et al. 2008 246 56 Adolescence
Journal of Applied
Bono et al. 2008 57 40 Psychology
Brown et al. 2007 245 56 PsychoMal Science
Journal of Experimental
DeHart et al. 2007 100 30 Social Psychology
Fleeson 2007 26 56 Journal of Personality
Graham 2008 38 49 Journal of Personality and
Social Psychology
Hogarth et al. 2007 74 60
Risk Analysis
Ilies et al. 2007 106 30 Journal of Applied
Psychology
Impett et al. 2008 55 14 Journal of Personality and
Social Psychology
Jones et al. 2007 420 29 Journal of Applied
Psychology
Kane et al. 2007 124 56
Psychological Science
Kubiak et al. 2008 16 28 Appetite
Kuppens 2007 80 59 Journal of Research in
Personality
Lucas et al. 2008 144 52
Journal of Personality
Moberly & Watkins 2008 108 56 Journal of Abnormal
Psychology
Moghaddam & Ferguson 2007 70 31
Journal of Personality
Nezlek et al. 2008 36 126
Emotion
Oishi et al. 2007 332 21 Journal of Personality and
Social Psychology
Piasecki et al. 2007 50 14 Journal Of Addictive
Behaviors

 

 

 

 

 

 

Table 2 (cont’d).

 

 

 

 

 

 

Journal of Research on
Schneiders et al. 2007 131 45 Adolescence
Snir & Zohar 2008 65 28 Applied Psychology: An
International Review
Song et al. 2008 230 41 Journal Of Applied
Psychology
Summerville & Roese 2007 34 98 Journal Of Experimental
Social Psychology
Thewissen et al. 2008 154 60 Journal Of Abnormal
Psychology

 

 

 

 

 

 

Note. N = Sample Size; T = series length.

Along with the many benefits gained from collecting longer series for each
individual comes increased methodological complexity. With the increase in series
length, longitudinal data structures are becoming intensive longitudinal designs (Walls &
Schafer, 2006). As this happens psychological longitudinal data starts to mirror the
structure and properties of time series that can be seen in biomedical and economic
literature. In medicine, short time series are considered to have as few as eight time
points (Ernst, Nau &, Bar-Joesph, 2005). In economics, short time series typically have
between only five and nine time points (Bhargava & Sargan, 1983; Hsiao, Pesaran, &
Tahmiscioglu, 2002). With event sampling, psychological data structures can have
anywhere from thirteen to over one hundred time points (from Table 2), well within the
range of being considered time series. It is important to note that with this increase in
series length, comes an increase in the complexity of the components of the data that
need to be addressed by statistical models (Walls & Schafer, 2006). It is easy to ignore
this increased complexity since the statistical packages psychological researchers use,
such as HLM, can easily handle the increased number of time points. However, by

ignoring the methodological issues that arise from having longer series, researchers can

get an inaccurate representation of the underlying psychological relationships. Therefore,
it is important for psychological researchers to consider and account for the increased
complexity that comes from dealing with longer time series. More specifically, it is
important to understand, identify, and deal with stochastic processes and their effect on

spurious regression.

Longitudinal Data Analysis

The primary issue dealt with in psychological longitudinal data analysis is the
existence of heterogeneous response processes that result in clustered data. In
longitudinal data structures it is common to observe heterogeneity across individuals on
the psychological phenomena of interest (Collins & Horn, 1991; Collins & Sayer, 2001;
Harris, 1963; Moskowitz & Hershberger, 2002). Heterogeneity that goes unnoticed or
unmodeled can lead to biased results and faulty inferences (Barcikowski, 1981; Krefl &
De Leeuw, 1998; Molenaar, 2004; Winer, Brown, & Michels, 1991). Therefore, it is
important to account for the presence of clustering due to heterogeneous response
processes in the data. The presence of clustering leads to three possible analytical
techniques that psychological researchers can use: pooled, disaggregated, or partial
poohng.

For most of the previous century, researchers utilized repeated-measures analysis
of variance (RM-ANOVA) to analyze longitudinal data. This technique pools all of the
data together and computes the average growth across people (Gelman & Hill, 2006;
Winer, et al., 1991). This method has a number of flaws. First, by pooling the data all

variability in the sample is lost. Thus, it becomes impossible to study or predict

individual differences in growth processes. The analysis only describes the average
grth across people and may not describe any one individual’s actual grth process.
The second flaw, and possibly the most important, is that pooling the data ignores any
clustering that may exist in the data. Pooling the data treats every observation as
independent. The presence of clustering indicates that all data points are not independent
and thus should be grouped together. When treating clustered data as independent RM-
ANOVA analyzes the data with significance tests that are too liberal due to the fact that
the sample size is considered larger than it should be. This leads to the results being
biased and an inflation of Type I error rates. In fact, even in situations where only a
moderate degree of clustering exists, the Type I error rate can increase to as high as 70%
when the number of individuals in each group is large (e.g., N = 100) (Barcikowski,
1981; Kreft & De Leeuw, 1998).

A second possibility when analyzing heterogeneous longitudinal data is to
completely disaggregate the data by analyzing each group or person separately and
calculating individual growth curves. This method has flaws as well. Groups or
individuals with few data points are weighted as strongly as those with many data points.
This is ill-advised because with less information the reliability of the findings decreases.
This problem could potentially be compounded if outliers exist in the groups with fewer
data points. Outliers would influence groups with small numbers of data points more
strongly than the overall sample or groups with more data points thus overemphasizing

their impact. This could lead to inaccurate conclusions or inferences (Gelman & Hill,

2006)

10

The final method for analyzing longitudinal data is random coefficient modeling
(RCM). RCM attenuates the problems of the prior two methods by compromising and
partially pooling the data in an attempt to accurately represent any heterogeneity that
exists between groups. The degree to which each groups’ data gets pooled is a function of
how much variability exists within the group and the amount of information available
(number of data points) for that group. Groups that have a lot of within-group variance or
very few data points get pooled in an attempt to bring the data closer to the mean. On the
other hand, groups with strong within group agreement or large amount of information
hardly get pooled at all (Gelman & Hill, 2006). This partial pooling allows for greater
reliability for each group and, more importantly, eliminates the inflation of Type I errors
due to clustering (Kreft & De Leeuw, 1998).

Along with the fact that RCM eliminates the problem of inflated Type I error rates
due to clustering in the data, it has a number of additional advantages over RM-ANOVA
and other analytical methods for longitudinal data. For example, RCM can easily handle
missing data or data that is unevenly spaced between participants and can allow for the
addition of continuous variables as predictors. Another advantage of RCM over previous
analytical methods is that RCM allows for error structures that are both correlated and
heteroscadastic. In longitudinal studies it is possible for the reliability of the data
collected to change over time. Likewise it is possible that errors among individuals or
groups across time points will become correlated (e.g., common method bias). Therefore,
it is assumed in RCM that errors are correlated and heteroscadastic within groups or
individuals but not between groups or individuals. This leads to a more complex error

structure known as the block diagonal error structure (Gelman & Hill, 2006; Singer &

11

Willet, 2003). This is an advancement over RM-ANOVA where one of the primary
assumptions is that errors are uncorrelated and have equal variance (sphericity) (Winer et
al., 1991). The assumptions about the underlying error structure in RCM allow for a more
accurate representation of the error processes that are commonly Observed in longitudinal
data than those of RM-ANOVA. All of these advantages resulted in RCM being the
primary analytical tool used by psychological researchers when dealing with longitudinal

data.

The Problem Presented

While RCM does provide a good solution for dealing with clustered data, it does
not account for the methodological concern of spurious regression that can result from
the presence of stochastic trends in the data that becomes more influential when
analyzing longer time series. The regression model and generalizations of that model,
such as RCM, require the assumption that the dependent variable is the result of a purely
deterministic process. This is particularly problematic because frequently the focus of
psychological longitudinal studies is on growth processes where stochastic processes
have been observed in many other fields of study (e. g., computer science, physics,
genetics, and economics). When the regression model is used to analyze stochastic data a
serious inflation of Type I error rates, called spurious regression, is frequently observed.
Spurious results are likely to be obtained anytime one stochastic series is regressed on
another, a stochastic series is regressed on time, and when a stochastic series is regressed

on time and other stochastic predictors (Granger & Newbold, 1974; Nelson & Kang,

1984).

12

To make matters worse, in the presence of stochastic trends as the number of time
points increases, the significance tests used in regression analyses diverge, resulting in
even greater Type I error rates (Durlauf & Phillips, 1988; Phillips, 1986, 1987).
Therefore, the increased focus on the study of growth processes combined with the longer
series used to study these processes leaves psychological researchers extremely
susceptible to spurious regression when using RCM or other generalizations of the
regression model. Two of the most dangerous cases are when Time or Level 1, time-
varying covariates are in the model as predictors since both of these types of variables
were shown to lead to spurious regression in the simple regression case (Granger &
Newbold, 1974; Nelson & Kang, 1984). Unfortunately, these cases frequently exist in
random coefficient modeling because a variable representing time is used in virtually all
RCM analyses during the initial model testing of the unconditional growth model and
since the inclusion of time-varying predictors is quite common and seen as a large
advantage of using RCM as a method of analyzing longitudinal data (Raudenbush &

Bryk, 2002; Singer & Willet, 2003).

Random Walks and Stochastic Processes
Random walks are one of the most commonly encountered and studied stochastic
processes. They are prevalent in virtually every scientific discipline including computer
science models of information search (Tang, J in, & Zhang, 2008) , physics models of
Brownian motion (Uhlenbeck & Omstein, 1930), genetic models of genetic drift (Wright,
1931), ecological models of biodiffusion (Skellam, 1951) and population dynamics

(Wang & Getz, 2007), and economic models of real GNP and employment (Nelson &

13

Plosser, 1982). In psychology, random walks are fundamental to the study of neuronal
firing (Gerstein & Mandelbrot, 1964), speeded categorization (Nosofsky & Palmeri,
1997), diffusion models of decision processes (Busemeyer & Townsend, 1993), and
consumer behavior such as new product adoption (Eliashberg & Chatterjee, 1986). The

simplest form of a random walk is described by the equation:

_ 1
Yt — Yt_1+et. ( )
Where Yt is the value at time t, Yt—l is the value at time t-l , and 6 t is a random error

term from a normal distribution that has a mean of zero and a constant variance, 0' e? As

seen in the equation, in random walks, each value is derived from the value of the data
point directly preceding it in addition to some random error term (Enders, 1995). A
random walk can be thought of as a series that is created by taking successive random
steps. Therefore, any point in a random walk time series is the accumulation of random
changes (i.e. error). Due to the fact that random walks are simply an accumulation of
error terms, it is impossible to predict future values. This means that the best guess or

expected value for any point in the future is the value directly preceding it. Therefore, the

initial condition (i.e. intercept), Y0, becomes the expected value for all fiJture time points,
as can be seen in the following equation:
E(Yt) = E(Yt—1)= E(Yo)- (2)

Since the intercept becomes the expected value for all time points, the mean of a random

walk is time-invariant. The variance of a random walk, however, follows the equation:

Var(Yt )= var(€t+ Et—1+ €1)=10'£2. (3)

14

Therefore, as time increases so does the variance, meaning that the variance at one time
point is not equal to the variance at any other time point. Figure 1 shows a random walk

that appears to be a negatively directed growth process.

 

 

 

 

Time

Figure 1. Random Walk

It is possible for random walks to be a bit more complicated and incorporate an
environmental factor in the form of a drift or time trend. The distinction between
stochastic and deterministic trends is seen most easily by examining the equation for a

random walk with drift,

_ t
Yt - Y0 + [it + 2i=1 6t. (4)
Yt is the current value of a variable, Y0 is the initial condition of a variable, fit is the

deterministic component of the series, called a drift, and 2L1 6 t is the stochastic

15

component of the series representing an accumulation of random errors. This equation

looks remarkably similar to the purely deterministic equation that is used in regression
(Yt =0 + [31 + Q). Ofien, it is very difficult to distinguish between series that are
purely deterministic and ones that are purely stochastic. Figure 2 plots a regression line
on data generated by either a purely deterministic or purely stochastic trend. Without the

labels it would be near impossible to tell them apart, yet it is imperative that scientists are

able to do so to make correct statistical and scientific inferences.

Deterministic Trend Stochastic Trend

 

Response

 

 

 

 

 

 

 

Time Time

Figure 2. Deterministic and Stochastic Trends

16

For the purposes of this thesis, only the simplest case of random walks will be
considered. Therefore, the effects of drifts and trends will be ignored. The addition of
either a drift or trend would further complicate matters by making the random walk look
even more like typical growth data than the stochastic trend demonstrated in Figure 2 and
would result in a significant fixed effect for the slope of Time indicating that on average
there is growth.

The prevalence of stochastic processes in the psychological sciences is still
unknown. There are a number of potential areas of research that could incorporate
stochastic trends in longitudinal data. One such area in organizational psychology is in
the study of mood and emotions. Affective Events Theory is the dominant approach to
the study of affect. It states that mood and emotions fluctuate over time and that events
in the environment are proximal causes for affective reactions (Weiss & Cropanzano,
1996). Since environmental events occur in a somewhat random or stochastic fashion, it
is possible that people’s reactions to those events follow a somewhat random path
(stochastic trend). Similarly, since almost all affect research relies on self report data, it
is possible that over time measurement error is being compounded, which would lead to
an accumulation of errors resulting in a stochastic process. Affect is just one area of
research that has the potential to be impacted by the presence of stochastic trends in
longitudinal data. Further investigation is needed to indentify the structure of affective

data and to identify other potential stochastic processes in psychology.

17

Spurious Regression
Granger and Newbold (1974) first documented spurious regression in economics.
They observed that frequently in the economic literature when regression was used to

determine the relationship between time series, the regression coefficients were almost

always significant and the amount of variance explained (R 2) was extremely high. They

suggested this result was due to the presence of stochastic trends in the data, leading the
regression models to inaccurately overestimate the true relationship between series. They
demonstrated this problem by regressing one independent random walk on another. Since
independent random walks are created only by an accumulation of errors, no true
predictive relationship can exist between them. Therefore, any significant results beyond

the nominal rate are due to problems with the estimation method. Granger and Newbold

found that when regression is applied to random walks the variance of R 2 becomes too
large, causing the distribution to no longer be unimodal around the origin. This results in

2 . . .
R being conSIstently overestimated, where the expected value goes from zero to 0.47

(47% of the variance explained). They also found that the standard error of the regression
coefficient is grossly underestimated leading to a significance test that is too liberal. This

results in the regression coefficient being significant approximately 76% of the time (for
a = 0.05), well beyond the nominal rate. To make matters worse, two unrelated series can

often appear to covary, thus making their true relationship harder to detect. This has been

documented many time using random walks; an example of such a case can be seen in

Figure 3.

18

 

 

 

 

Time
Figure 3. Independent Random Walks
Nelson and Kang ( 1984) expanded on the findings of Granger and Newbold
(1974). They looked at how spurious regression manifested in the presence of a purely
stochastic series (random walk) being regressed on to both an independent stochastic
series (random walk) and a deterministic variable, Time. Just like in the previous case, the
dependent random walk was created only from the accumulation of errors, so all

significant results for any predictor beyond the nominal rate are due to faulty estimation.

R 2 can never decrease as a result of adding predictors to the model so it is not surprising

19

that the addition of Time as a predictor increased R 2 from 0.47 in the prior case to 0.50.

The standard error of both Time and the predictor random walk were heavily
underestimated resulting in very liberal significance tests. The addition of Time as a

predictor decreases the percent of times that the slope parameter for the predictor random
walk is significant from 76% in the prior case to 64% (for a = 0.05). The slope of Time,

on the other hand, is significant 83% of the time; meaning that once again both predictors
are significant well above the nominal rate. To make matters worse, the effects of
spurious regression are exacerbated as the number time points increases or the number of
additional stochastic predictors increases (Granger & Newbold, 1974; Nelson & Kang,
1984; Phillips, 1986, 1987).

The effects of spurious regression are quite dramatic and could lead to many
incorrect results and inferences if they are not accounted for. To show how spurious
regression could play out in random coefficient models, a conceptual and mathematical
overview of RCM will first be explained, followed by integrating the effects of spurious

regression due to stochastic trends in the data with the models used in RCM analyses.

RCM Conceptual and Mathematical Overview
Random coefficient modeling (RCM) allows researchers to analyze longitudinal
data using a two step process. Initially, the focus is on determining the shape and
trajectory of grth curves and analyzing the degree to which there is heterogeneity in
the response process. This initial step is done by partially pooling the data according to
the within-group variance and amount of information available for each group. Then,

RCM uses maximum likelihood estimation to estimate the hyperparameters for each

20

variable (slope and intercept) in the model to maximize the fit between the observed and
expected variance—covariance matrices. Hyperparameters are the mean and variance of
each parameter across all groups or individuals in the sample. From these estimates the
total within- and between-group variance is analyzed using the intraclass correlation
coefficient (1C C( I )). This is where the second step in the data analysis process begins. If
a significant amount of variance (either within or between) is left unexplained, RCM
allows researchers to add predictor variables to the model in an attempt to explain
additional variance. These predictors can take the form of either Level 1, within-person,
predictors or Level 2, between-person, predictors. There are two distinct types of Level 1
predictors: time-invariant and time-varying. Time-invariant predictors vary between
people but take on the same value for all time points within a given person. Time-varying
predictors vary both between and within individuals over time. To evaluate the fit
between models a log-likelihood ratio test or chi-squared difference test (called the
deviance statistic) is used. The goal, as in all model comparisons, is to find a model that
explains a maximum amount of variance while being as parsimonious as possible.

When psychologists utilize RCM to analyze their longitudinal data, the
recommended practice is to begin by specifying two models as baseline models for future
analyses. These two models are the multilevel unconditional means model and the
multilevel unconditional grth model (Singer & Willet, 2003). These models are used
to determine the trajectories of the growth curves and analyze the variance components.

The multilevel unconditional means model, as represented by Singer and Willet

(2003), is specified as:

Yij = "0: + Eij (5)

21

”Oi = 700 + {ija

where it is assumed that:

2 2
51'1”” N“), as ) and Ci}: ~ MO, 00 )-
Yi j is the dependent variable measured for each person i and each occasion j, 1t0i is the
mean of Y for individual i, 700 is the mean of Y across everyone in the population, 0' 52

is the pooled variance of each individual’s data around his/her mean, and 0'3 is the

pooled variance of individual-specific means around the grand mean.

The unconditional means model splits the total variance into within- and between-
person variance components. Using these variance components the intraclass correlation
coefficient (ICC (1)) is computed, which indicates the proportion of total variance
residing between individuals. From this, one can evaluate whether a substantial amount
of variance resides within and between individuals. If it is determined that there is a
significant amount of within- and between-person variance then it is suggested that the
researcher search for predictors to explain both the within- and between-person variance.
However, before adding substantive predictors it is recommended to run an additional
baseline model, the multilevel unconditional growth model, to determine the shape and
trajectory of the growth curves.

The multilevel unconditional growth model, as represented by Singer and Willet

(2003), is specified as:

22

Yij = ”Oi + fill-Time”- + Eij
”01 = 700 + (0i (6)

7T1t = 710 + C1D

where it is assumed that:

~,N(0 03)) and [ZOi]~N (3] [2): 001211)

Yii is the dependent variable measured for each person i and each occasion j, 1! 0i is the
initial status of Y for individual i, 700 is the fixed effect for the intercepts and is the

initial status of Y across everyone in the population, 11' It is the rate of change of Y for

individual i, 710 is the fixed effect for the slopes and is the rate of change of Y across

everyone in the population, 0' E is the pooled variance of each individual’s data around
their linear change trajectory, 0'3 is the variance component of the random effect of the

intercepts and is the unpredicted variability in initial status, 0'12 is variance component of

the random effect of the slopes and is the unpredicted variability in rate of change, and

0'10 is the population covariance between intercepts and slopes.

A log-likelihood ratio test allows the researcher to evaluate whether the
unconditional growth model fits the data better than the unconditional means model. If
this statistic is significant, the researcher would conclude that the unconditional growth

23

model is a better representation of the data than the unconditional means model. If the
resulting model still has a large amount of variance to be explained, in either the initial
status (intercept) or the rate of change (slope), it is recommended that investigators search
for additional predictors to explain the heterogeneity (Snijders & Bosker, 1999). Singer
and Willet (2003) echo this notion by saying that researchers need to look at the variance
components to “assess whether there is hope for future analyses” (p. 99).

If the researcher wishes to explain additional variance there are many possibilities
that he/she can try and it is recommended that theory guide the choice of all subsequent
predictors. When using RCM, additional predictors can take on two forms: Level 1
predictors and Level 2 predictors. Level 1 predictors are those that vary within
individuals or groups. Some examples are salary, fatigue, intelligence, and mood. Level 2
predictors are those that vary between groups or individuals but are constant within
groups or individuals. These are often grouping variable such as gender, race, religion,
etc. Both Level 1 and Level 2 predictors can be entered into the model as a fixed effect, a
random effect, or both. Predictors are usually added as fixed effects first and then the log-
likelihood ratio test is used to determine whether additional variance is explained. Then,
if there is still a significant amount of variance left unexplained, the predictor is allowed
to vary as a random effect (Singer & Willet, 2003).

There are two distinct types of level 1 predictors: time-invariant and time-varying.
Level I, time-invariant predictors are things like salary, intelligence, personality, etc.
where the variables can vary between people but are constant within a person over time.
Level 1 time-varying predictors are things like fatigue, mood, etc. that can vary not only

between people but also can vary within a person over time. Singer and Willet (2003)

24

state that “because each predictor — whether time-invariant or time-varying — has its own
value on each person’s multiple records. A time-invariant predictor’s values remain
constant; a time-varying predictor’s values vary. There is nothing more complicated to it
than that” (pg. 160). Statistically speaking from an analytic point of view, this is
absolutely true; however, inferentially there are great differences between time-invariant

and time-varying predictors.

Spurious Regression in RCM

The current literature on the phenomenon of spurious regression only focuses on
the simple regression case that looks at the aggregate relationship between variables
across people. In random coefficient models panel data are collected and analyzed using
separate series for each participant in the sample. To estimate how the effects of spurious
regression would play out in grth curve analysis with panel data, one must consider
the distribution theory for the parameter estimates of the simple regression model in the
presence of stochastic trends in the data. Durlauf and Phillips (1988) derived the expected
values and distributional convergence properties for each component of the regression
model for the case of spurious regression due to stochastic data. With respect to RCM,
the expected means of the distribution for each regression parameter represent the
expected value for the fixed effects in RCM. Likewise, the expected variances of the
distribution for each regression parameter represent the expected value for the variance
components of the random effects in RCM. These insights allow for predictions to be

made on how the various parameters in RCM will behave in the presence of stochastic

data.

25

The application of the findings of Durlauf and Phillips (1988) and Nelson & Kang
(1984) will be discussed for the fixed effects, the significance tests on the fixed effects,
the variance components for the random effects, the significance tests on variance

components for the random effects, and the deviance statistics.

Fixed Effects

Durlauf and Phillips (1988) mathematically proved that the average effect for
both the intercept and slope of Time is zero. Nelson and Kang (1984) provide support for
these findings by providing Monte Carlo simulations demonstrating that on average
across replications the effect of the intercept and slope of Time is empirically zero.
Therefore, since the fixed effects are estimated by calculating the average effects of each
parameter across people, the fixed effects for both the intercept and slope of Time should
be approximately zero for all models analyzed, regardless of the number of time points.
When other stochastic predictors are added to the model they have been shown to
converge weakly to a value which has empirically been demonstrated to be zero on
average (Durlauf & Phillips, 1988; Nelson & Kang, 1984). Consequently, the fixed
effects for all additional stochastic predictors should be approximately zero, regardless of

the number of time points.

Significgice Tests on the Fixed Effects
While the fixed effects across all models for all parameters are on average
approximately zero, the behavior of their significance tests varies a great deal depending

on what other parameters are in the model. When parameters are added to the model as

26

only a fixed effect, it is essentially like adding them to the standard regression model.
Therefore, all results obtained from previous research on simple spurious regression
should be observed in the multilevel model as well. The significance tests on the
parameter estimates for simple regression were proven to diverge with increased time
points. This happens because as the number of time points increases, the regression
model increasingly underestimates the standard errors of the parameters relative to their
standard deviations, resulting in significance tests that are too liberal. This should cause
the rejection rates for the parameters entered only as fixed effects to increase as the
number of time points increases to well above the nominal rate (Durlauf & Phillips, 1988;
Nelson & Kang, 1984).

When predictors are added to the model as both fixed and random effects the
significance tests for the fixed effects are much better behaved. As demonstrated in
Appendix A, when variables are entered as both fixed and random effects the standard
errors of the fixed effects closely approximate the standard deviations of those effects
across the reasonable range of time points studied in psychology. Therefore, the t-ratios
for the fixed effects are accurately estimated leading the rejection rates on those fixed

effects to stay roughly nominal.

Variance Components for the Rgdom Effects
The variance components for the random effects for each parameter in all models
are equivalent to the variance components of the distributions derived by Durlauf and

Phillips (1988). The variance of the intercept parameter increases linearly as a function of

the number of time points defined by the equation '1—5. The variance of the parameter

27

estimate on the slope of Time, however, converges toward its true value of zero as a

6
function of the number of time points defined by the equation 371:. Nelson and Kang

(1984) once again support these findings by empirically demonstrating the effect of
number of time points on the variance of the parameters of both Time and the intercept
using Monte Carlo simulations. As the number of time points increases, the variance
component of the random effect of the intercept should tend toward infinity while the
variance component for the random effect for the slope of Time should converge toward
its true value of zero. When additional predictors are added to the model as random
effects the total variance of the system is divided up further. This will cause the above
relationships between the number of time points and the variance components for the
random effects to be weakened; however, their distributional convergence properties will
remain the same. The variance of an additional predictor, X, was shown to converge
weakly to a distribution, meaning that the variance of X will converge towards a value
and remain at approximately that value for all additional numbers of time points. As a
result, the variance component for the random effect for X should be approximately a

constant for all reasonably large number of time points (Durlauf & Phillips, 1988).

_S_ignificg1ce Tests on the Variance Components of the Rgndom Effects

To determine how the significance tests for the variance components for the

random effects will behave the equation used to estimate the [2 statistic must be

analyzed. Raudenbush and Bryk (2002, Equation 3.103) define the equation as:

28

EMU—7610’}; yquSJ'Y

qui

 

(7)

9

where qu j is a block diagonal covariance matrix of errors with each block defined as:

v-=az(X,-TX,-)'1. (8)

By combining Equations 7 and 8 we get:

mfg—7110‘: 7qswsi)2

(02(XTXI)-1) . (9)

1'1'

 

The denominator of Equation 9 gets small incredibly fast as the number of time points

. T ‘1 . . . .
Increases because (X1- Xj) becomes exponentIally larger With each Increased time

point. This occurs because each additional time point results in the X j matrix increasing

by a factor of N (because each block increases by one and there are N blocks). Also, not
only does the matrix greatly increase in size with the addition of each time point, but the

variance of Time increases as the number of time points increases as well. This leads to

the value of X j greatly increasing with every additional time point. The relationship

between 0' 2 and number of time points is linear and does increase with increased time

points, but the magnitude of the increase is significantly smaller than that of the X j

matrix. Therefore, the denominator gets extremely small at a rapid pace. The numerator

also gets smaller as the number of time points increases but at much slower rate than the

29

denominator. This causes the value of the [2 statistic to go towards infinity as the

number of time points increases which results in the tests on the random effects being

significant. Adding more predictors to the model does not change the relationship

described above. By adding in a stochastic predictor, X, the X 1- matrix will increase even

more with the addition of each time point because the matrix gets larger by a factor of 2N
(each block increases by 2 due to addition of another time point and another value of X).
As a result, the significance tests for all the variance components for the random effects

should be significant.

Devignce Statistic

The final statistic that is analyzed is that of the 12 test between models. The

behavior of the Z2 test is completely predictable because it is a combination of the
significance of each additional predictor added to the model. Consequently, when the
difference between models is only the addition of a fixed effect, the 152 test will be

significant the same proportion of times that the significance test on the fixed effect is
significant. Since when stochastic predictors are only added as fixed effects their

significance tests become significant more frequently with increased number of time
points, so will the 12 test. When predictors are added to the model as random effects the

significance tests on the variance components of those effects will be significant.

Therefore, the [2 test between the models will also be significant.

30

As mentioned earlier, RCM is typically used to analyze longitudinal data through
a stepwise process of nested model testing (Kreft & De Leeuw, 1998; Raudenbush &
Bryk, 2002; Singer & Willet, 2003; Snijders & Boskers, 1999). Six common models will
be explained in this research. The first model presented will be the unconditional means
model. The second model will be the unconditional grth model followed by model
three which is the unconditional grth model with X included as a fixed effect. The
fourth model will build fi'om model three and have X as both a fixed and random effect.
Some researchers choose not to include Time in the model after the initial model testing
stage. As a result, a fifth model will be discussed that is the unconditional means model
with X included as a fixed effect. The final model will build on model five by includeding
X as both a fixed and random effect. Each model will be presented, the model parameters
will be briefly explained, and then hypotheses regarding each parameter estimate and
their corresponding significance test will be made using the above rationale. The model

notation used will be taken from Singer & Willet (2003).

Model 1: Unconditional Megs Model

The first recommended model is the unconditional means model:

Yij =7T0i + Eij
(10)
”Oi = 9,00 +6179

where it is assumed that:

Eij N N10, 03) and (i1 ~ N(0, Uri)-

31

This model includes a fixed effect for the intercept, 700, a random effect for the

intercept, (0i, a variance component for the random effect of the intercept, 0'3 , and a

variance component for the residual error terma'ez. The primary purpose in running the

unconditional means model is to calculate the ICC(1) to determine the proportion of

variance that exists between groups or individuals. The [C C ( I ) is calculated by the
amount of variance between individuals, 002 , divided by the total variance (0'62 + 0' g ).

For a random walk, the within person (residual) variance is typically set to one. The
between person variance or the variance between the means across random walks is equal
to the variance of the intercepts. This occurs because the expected value or mean for a

random walk is its initial value (see Equation 2). The variance at any time point is equal

to 0' 2t (Equation 3) and the intercept is calculated at t = 1 so the between-person

variance is also equal to one. Therefore, the true [C C ( I ) for random walks is equal to

 

(1+1) or 0.5. When the regression model is used to analyze stochastic data (random

. . . . . . . T+1
walks) the estimated variance between IndIVIduals Is apprOXImately —3- 0' E , compared

. . . . . . . . . . “”1
to the estimated variance WIthIn IndIVIduals which Is approx1mately —6- 0'3 (Nelson &

Kang, 1984). Using these values, the expected ICC (I) for random walk data is:

 

T+1 2
T06 * 2
ICC(I )= "771 T+1_2_=§- (11)

2
3 UE*+-—6—O'€*

32

Consequently, the 1CC(1) should overestimate the true amount of variance that
exists between individuals with a value of approximately 0.66.

Hypothesis 1: The ICC ( I ) obtained from the unconditional means model will be

overestimated and indicate approximately 66% of the variance exists between

individuals.

Model 2: Unconditional Growth Model
After determining that a significant amount of variance exists between individuals
the next recommended step in RCM is to run the unconditional grth model specified

as:
Y1; = [700 + 710Timeijl+ [Cot + (uTimet-j + 6:11 (12)
This model includes fixed effects for the intercept and slope of Time, 700 and 710

respectively, as well as random effects for the intercept and Time, {0 i and (1i
respectively, along with variance components for the random effects for the intercept and

. 2 . . .
T Ime, 0'0 and 0'12 respectively, and a vanance component for the reSIdual error term,

0'3. Following the logic and findings presented above, the hypotheses for the parameter

estimates and significance tests for Model 2 are as follows:
Hypothesis Za: T he fixed eflect of the intercept will be approximately zero
regardless of the number of time points.
Hypothesis 2b: The fixed effect of the slope of Time will be approximately zero

regardless of the number of time points.

33

Hypothesis 3a: The significant test for the fixed effect of the intercept will stay
approximately at the nominal rate with increased time points.

Hypothesis 3b: The significant test for the fixed eflect of the slope of Time will
stay approximately at the nominal rate with increased time points.

Hypothesis 4a: The variance component for the random effect of the intercept will

diverge toward infinity as the number of time points increase. The empirical value

2T

for any given number of time points will be approximately '1—5'.

Hypothesis 4b: The variance component for the random effect of the slope of Time

will converge toward zero as the number of time points increase. The empirical

value for any given number of time points will be approximately S—T'

Hypothesis 5a: The significance test on the variance component for the random
efl'ect for the intercept will be significant regardless of the number of time points.
Hypothesis 5 b: The significance test on the variance component for the random

effect for the slope of Time will be significant regardless of the number of time

points.
Hypothesis 6: The 12 significance test (also called the deviance statistic) will

indicate that the unconditional growth model fits the data better than the

unconditional means model.

34

Model 3: Unconditionfl Growth Model with X as L F ixed Effect

After determining that the unconditional growth model fits the data better it is
common for psychological researchers to add in predictors to explain any remaining
unexplained within- or between-person variance. It is suggested by methodologists to
first input the predictor variables into the model as only fixed effects (Singer & Willet,

2003). The predictor, X, in this model is entered as a Level I, time-varying covariate.

This model has the form:

Yii = [700 + hammer} + 720Xij] + [Cor +
(13)
(ll-Time”- + EU]-

This model includes fixed effects for the intercept, slope of Time, and X, 700, 710, and

720 respectively, as well as random effects for the intercept and Time, (Of and (1i

respectively, along with variance components for the random effects for the intercept and

Time, 0'3 and 0'12 respectively, and a variance component for the residual error term,

0'62. The hypotheses for Model 3 resulting from the expectations derived above are:

Hypothesis 7 : The parameter estimates for the fixed effects and the variance
components of the random eflects for the slope of Time and the intercept will be
approximately the same as for the unconditional growth model.

Hypothesis 8: The significance tests for the fixed effects and the variance
components of the random effects for the slope of Time and the intercept will

follow the same patterns as for the unconditional growth model.

35

Hypothesis 9: T he fixed effect for X will be approximately zero, regardless of the
number of time points.

Hypothesis 10: The significance test for the fixed effect of X will diverge as the
number of time points increases resulting in X being significant more frequently

for higher numbers of time points.
Hypothesis 11: The 12 significance test between the unconditional growth model

and the model with X as a fixed effect will be equivalent to the significance test on
the fixed effect parameter for X and thus diverge as the number of time points

increases resulting in the test being significant more frequently for higher

numbers of time points.

Model 4: Unconditional Growth Model with X as a Fixed grid Random Effect
After adding in X as only a fixed effect, the next recommended step is to allow X

to vary and enter it into the model as a random effect as well. The model is:

Yii = [700 + 710Tlmeii + yzoXiil + [(01' +
(14)
Cunmeij + (zixij + 5:7]-
This model includes fixed effects for the intercept, slope of Time, and X, 700, 710, and

yzo respectively, as well as random effects for the intercept, Time, and X, (0i: (1i, and

(21-, respectively, along with variance components for the random effects for the

36

intercept and Time, 0'3 , 0'12 , and 0'22 respectively, and a variance component for the

residual error term, 0' g

Hypothesis 12: The parameter estimates for the fixed effects for the slope of Time
and the intercept will be approximately the same as for the unconditional growth
model and the model with X as only a fixed effect.

Hypothesis 1 3: The significance tests for the fixed efl'ects for the slope of Time
and the intercept will follow the same patterns as for the unconditional growth
model and the model with X as only a fixed eflect.

Hypothesis 14a: The variance component for the random eflect for the intercept

will diverge toward infinity as the number of time points increases. The empirical

value for any given number of time points will be slightly less than 1;.

Hypothesis 1 4 b: The variance component for the random effect for the slope of
Time will converge toward zero as the number of time points increases. The

empirical value for any given number of time points will be slightly greater than

6

5T.
Hypothesis 15: T he fixed effect of X will be approximately zero, regardless of
number of time points.

Hypothesis 1 6: The significant test for the fixed eflect of X will stay approximately

at the nominal rate with increased time points.

37

Hypothesis [7: The variance component for the random effect of X will converge
to a value and approximately stay at that value for all additional number of time

points.

Hypothesis 18: The significance tests on the variance components for the random

eflects for the intercept, slope of Time, and X will be significant.

Hypothesis 19: The ,1} significance test between the model with X as a random

effect and the model with X only as a fixed eflect will be significant.

Model 5: Unconditional Means Model with X as a Fixed Effect

When testing the impact of Level 1 time-varying covariates some researchers opt
to remove Time from the model after testing the unconditional grth model. While this
decision may be theoretically valid, it does not remove the effects of spurious regression.
Typically, what researchers who choose this route do is analyze the unconditional means
model to assess the ICC(1), then test the unconditional growth model to determine the
average effects and trajectory of the growth processes. Researchers would then generally
remove Time from the model and insert the predictor as only a fixed effect first, as was

done in the previous models. The model typically takes the form:
Yr} = [700 + VioXiil + [Cor + en]. (15)
This model includes fixed effects for the intercept and X, 700, and 710, respectively, as

well as a random effect for the intercept, (Oi, along with a variance component for the

random effect for the intercept, 0'3 , and a variance component for the residual error term,

38

0'62. This model is roughly equivalent to the third model presented that had Time as well

as X predicting the dependent variable. Therefore, the hypotheses follow closely in line
and are:
Hypothesis 20: T he fixed effect of the intercept will be approximately zero
regardless of the number of time points.
Hypothesis 2]: The significant test for the fixed eflect 0f the intercept will stay
approximately at the nominal rate with increased time points.
Hypothesis 22: The variance component for the random effect for the intercept
will diverge toward infinity as the number of time points increases. The empirical

2T
value for any given number of time points will be approximately TS.

Hypothesis 23 : The fixed effect of X will be approximately zero, regardless of
number of time points.
Hypothesis 24: The significance test for the fixed eflect of X will diverge as the

number of time points increases resulting in X being significant more frequently

for higher numbers of time points.
Hypothesis 25: The ,‘(2 significance test between the unconditional means model

and the model with X as a fixed eflect will be equivalent to the significance test on
the fixed effect parameter for X and thus diverge as the number of time points

increases resulting in the test being significant more frequently for higher

numbers of time points.

39

Model 6: Unconditional Meags Model with X as a Fixecfag Random Effect
After testing the addition of a predictor, X, as only a fixed effect it is common to

allow the predictor to vary as a random effect as well. This model takes the form:
Yij = [700 + 710Xij] + [Cot +C1iXij '1' Eijl- (16)
This model includes fixed effects for the intercept and X, yOO and 710 respectively, as

well as random effects for the intercept and X, (0i and (Ii respectively, along with
variance components for the random effects for the intercept and X, 0' g and 012

respectively, and a variance component for the residual error term, 062. The model is

very similar in structure to that of the unconditional grth model and thus the
hypotheses are:
Hypothesis 26: The fixed effect of the intercept will be approximately zero
regardless of the number of time points.
Hypothesis 27: The significant test for the fixed effect of the intercept will stay
approximately at the nominal rate with increased time points.
Hypothesis 28: The variance component for the random eflect for the intercept

will tend toward infinity as the number of time points increases. The empirical

value for any given number of time points will be '1—5.

Hypothesis 2 9: T he fixed effect of X will be approximately zero, regardless of
number of time points.
Hypothesis 3 0: The significance test for the fixed effect of X will stay

approximately at the nominal rate with increased time points.

40

Hypothesis 3]: The variance component for the random eflect for X will stay
approximately constant as the number of time points increases.

Hypothesis 32: The significance tests on the variance components for random
effects for the intercept and X will be significant.

Hypothesis 33 : The 12 significance test between the model with X as a random

effect and the model with X only as a fixed eflect will be significant.

 

41

I»!

 

 

 

 

METHOD

To demonstrate the effects of stochastic trends on random coefficient modeling,
Monte Carlo simulations with random walks was used. Since event/experience sampling
methodology is becoming increasingly common and is the domain where most longer
time series in psychology are observed, the simulation parameters mirrored those
commonly found in event sampling. The two parameters that were manipulated are the
sample size and the number of time points. Based on the reported review of event
sampling studies (Table 2) the average number of participants is approximately 123 with
a relatively large range from 26 up to 420 participants. While sample size is not
hypothesized to affect the results due to stochastic trends it is important to account for the
possibility that it could have an impact via statistical power. To simulate the range of
possible sample sizes two values were analyzed. To evaluate effects of smaller samples
the lower value for sample size was set at 30. To simulate larger samples that are more
commonly found in the literature, the larger value for sample size was set at 150. For
each sample size four different series lengths were simulated. The studies in Table 2
indicate that on average the number of time points observed is 47 with a range from 14 to
126 time points. To simulate this large range of potential lengths four values were
analyzed for each sample size. These values were 5, 10, 40, and 70 time points. These
values cover the range of commonly occurring series lengths in the literature as well as
allow trends in the data over time to be observed. The overall design has two sample size

conditions with four series length conditions for each sample size for a total of eight

conditions (2 x 4).

42

Data for each condition was generated and analyzed in the statistical program, R.
The dependent variable (Y) was be created by generating an independent random walk for
each “participant” in the sample. The values that make up the dependent random walk
were randomly sampled from a normal distribution with a mean of zero and a variance of
one. The Time variable was created by having a linear increase from one to the maximum
number of time points (5, 10, 40, or 70). To make the intercept value more interpretable,
the Time variable had one subtracted from each value so it began at zero. For models 3
through 6 the predictor variable (X) was also created by generating independent random
walks for each “participant” in the sample. The values that make up the predictor random
walk were also randomly sampled from a normal distribution with a mean of zero and a
variance of one. Each model was analyzed using the lmer function in the LME4 package
in R. All simulation conditions were replicated 1000 times. The results presented for the
fixed and random effects are the average values across the 1000 replications. The results
presented for the significance tests for the fixed and random effects as well as the
deviance statistics are the proportion of times across the 1000 replications that the null
hypothesis was rejected. Since the random walks are independent and generated by

taking successive random steps, no true predictive relationships can exist. Therefore, any
relationship above the nominal rate (a = 0.05) between the variables is solely due to

faulty estimation caused by spurious regression.

43

RESULTS

Model 1: Unconditional Means Model
When researchers analyze the unconditional means model it is common to
evaluate the fixed effect of the intercept to determine the grand mean. As seen in Table 3,
the estimate for the fixed effect for the intercept was approximately zero in all eight
conditions, thus being well behaved. Similarly, the rejection rate on the intercept fixed
effect was also well behaved and was approximately nominal across all conditions. The

unconditional means model is also often used to determine whether a significant amount

of variance resides between means, commonly computed via a 12 test. Regardless of

sample size, the between group variance increased as the number of time points

increases. Across all conditions, the Z2 statistic was significant in every replication

indicating significant between mean variance. The most common statistic that is analyzed
from the unconditional means model is the ICC ( I ), which determines the percentage of
variance that resides between groups or individuals. Consistent with hypothesis 1, the
ICC (I) was systematically overestimated at 0.66 in all conditions. These results would
lead researchers to incorrectly believe that 66% of the total variance resides between
individuals and that the between group variance was significant, thus encouraging

additional model testing.

Table 3. Model 1 - Unconditional Means Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150
T 5|10|40]70 5110140170
Fixed Effect Statistics
7’00
(5 ) -001 0.01 0.01 0.00 -0.01 0.00 0.01 -0.02
700 (0.28) Q36) (0.67) (0.89) (0.12) (0.17) 40.31) (0.39)
SEy
00 0.06 0.05 0.05 0.05 0.04 0.05 0.05 0.04
P706 0.27 0.35 0.66 0.87 0.12 0.16 0.30 0.40
Random Effect Statistics
—2
00
1.95 3.55 13.21 22.99 1.99 3.65 13.51 23.55
P 2
0'0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
52
6
1.01 1.83 6.82 11.87 1.00 1.83 6.82 11.86
ICC(I)
0.65 0.65 0.65 0.65 0.66 0.66 0.66 0.66

 

 

 

 

 

 

 

 

 

Note. N = Sample Size; T = series length; 7’00 = average estimate of grand mean; 5700:
standard deviation of grand mean; SE700= standard error of grand mean; P700 =

rejection rate of hypothesis test on grand mean; 53 = average variance of individual

means; P 0.3 = rejection rate of hypothesis test on variance of individual means; 0'? =

average within-person variance; [C C( 1 ) = intraclass correlation coefficient.

Model 2: Unconditional Growth Model
When the unconditional growth model is fit to longitudinal data, a number of
parameters are evaluated. As shown in Table 4, the standard errors for the fixed effects of
both time and intercepts were accurately estimated relative to their respective standard
deviations. Therefore, both fixed effects were accurately estimated to be approximately

zero, with nominal rejection rates for all conditions, thus supporting hypotheses 2 and 3.

45

Table 4. Model 2 - Unconditional Growth Model - Fixed Effect Statistics

 

N 30 150

 

T 5 10 40 70 5 10 40 70

 

7’00

S -0.01 0.01 0.01 0.00 -000 -000 -0.00 -0.01
(700) (0.20) (0.24) (0.44) (0.56) (0.09) (0.11) (0.19) (0.25)

 

SE

 

700 0.05 0.05 0.05 0.05 0.05 0.04 0.05 0.05
P

700 0.20 0.24 0.43 0.55 0.09 0.11 0.19 0.25

 

7’10
000 0.00 0.00 0.00 0.00 0.00 000 0.00
(5)10)

 

(0:09) (0.06) (0.03) (0.02) (0.04) (0.03) (0:01) (0.01)
SE,
10

0.05 0.06 0.05 0.05 0.05 0.06 0.05 0.05

 

P

110 0.09 0.06 0.03 0.02 0.04 0.03 0.01 0.01

 

 

 

 

 

 

 

 

 

 

 

Note. N = Sample Size; T = series length; 7’00 = average estimate of population intercept;

5700: standard deviation of population intercept; 55700: standard error of population
intercept; P700= rejection rate of hypothesis test on population intercept; 710 = average

estimate of population slope; 5710: standard deviation of population slope; SE),10 =

standard error of population slope; P710 = rejection rate of hypothesis test on population

slope.
Consistent with hypothesis 4a, the variance component of the random effect for
the intercept increased as the number of time points increases (Table 5). Regardless of

sample size, for any given number of time points the variance component of the intercept

. 27' .
was approx1mately 1-5'. The vanance component for the random effect of Time, however,

. . . . . 6
decreased wrth Increased number of time pornts at a rate of approxrmately a, thus

supporting hypothesis 4b. For both the intercept and Time, the variance components for
the random effects were significant in every replication across all eight conditions,

46

consistent with hypothesis 5. Not surprisingly, since the random effects were significant

across all conditions, the deviance statistic between the unconditional means model and

unconditional growth model was also significant in every replication of every condition,

thus confirming hypothesis 6. These results would mislead researchers to conclude that

there was significance heterogeneity in the growth process and ultimately to search for

predictors of the unexplained heterogeneity. Since in reality all data came from the same

data generating mechanism, no true heterogeneity exists and all significant predictors

would solely be due to Type I error.

Table 5. Model 2 - Unconditional Growth Model - Random Effect Statistics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150

T 5 10 40 70 5 10 40 70
—2

00 0.92 1.50 5.33 9.03 0.95 1.54 5.45 9.41
P 2

0'0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
—2

01 0.21 0.1 l 0.03 0.02 0.21 0.1 1 0.03 0.02
Poi 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
—2

001 0.20 0.00 -O.18 -0.20 0.17 -001 -0.19 -0.21
P031 0.14 0.06 0.19 0.21 0.34 0.06 0.64 0.75
-2

06 0.47 0.81 2.80 4.82 0.47 0.80 2.80 4.80
P

412 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

 

Note. N = Sample Size; T= series length; 502 = average variance of intercepts; P05 =

rejection rate of hypothesis test on variance of intercepts; 512 = average variance of

slopes; P012 = rejection rate of hypothesis test on variance of slopes; 0:621 = average

correlation between intercepts and slopes; P031 = rejection rate of hypothesis test on

correlation between intercepts and slopes; 562 = average within-person variance around

47

 

linear trajectories; P A 12 = rejection rate of the hypothesis test on the difference between

model fit of unconditional means and unconditional growth models.

Model 3: Unconditional Growth Model with X as a Fixed Effect

If researchers choose to continue with additional model testing the next
recommended step is to insert predictors as only fixed effects. Doing so did not affect the
fixed effects or the variance components of the random effects for the intercept and Time.
Likewise, the corresponding significance tests were approximately identical to those in
model 2. These findings support hypotheses 7 and 8.

As shown in Table 6, the fixed effect for the time-varying predictor, X, was well
behaved and was approximately zero, supporting hypothesis 9. Unlike the fixed effects of
Time and the intercept, the standard error for the fixed effect of X is underestimated

relative to its standard deviation. This results in the significance tests for the fixed effect
of X exceeding the nominal rate (at 01 = 0.05). Regardless of sample size, as the number

of time points increased the standard errors were increasingly underestimated. This
caused the significance test for the fixed effect of X to diverge, resulting in extremely
inflated Type I error rates, consistent with hypothesis 10. Since the deviance test between
this model and the unconditional growth model is a one degree of fi'eedom test, it is
approximately equal to the rejection rate of the fixed effect of X. Therefore, as the
number of time points increases, the deviance statistic becomes significant a greater
proportion of times (Table 7). Since no true predictive relationship exists between the
criterion and the time-varying predictor, X, researchers are increasingly likely to make

incorrect inferences as the number of time points collected increases. This is counter

48

intuitive to most cases in psychology where having a greater number of time points is

preferred because it provides researchers with more information and gives greater

statistical power to detect true relationships. These findings would once again encourage

researchers to continue on to test additional models despite the fact that all findings,

current and future, are completely spurious.

Table 6. Model 3 - UGM with X as a Fixed Effect - Fixed Effect Statistics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150
T 5 10 40 70 5 10 40 70
700
(S ) -0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00
700 (0.20) (0.24 (0.44) (0.56) (0.09) (0.1 I ) (0.19) (0.25)
SEYQQ 0.05 0.05 0.05 0.04 0.05 0.04 0.05 0.05
P700 0.20 0.24 0.43 0.55 0.09 0.11 0.19 0.25
7’1 0
(S ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
710 (0.09) Q06) (0.03) (0.02) (0.04) (0.03) (0.01) (0.01)
S E 710 0.06 0.05 0.05 0.05 0.04 0.06 0.05 0.05
P710 0.09 0.06 0.03 0.02 0.04 0.03 0.01 0.01
720
(S ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
720 (0.09) (0.08) (0.08) (0.08) (0.04) (0.04) (0.04) (0.04)
SE7” 0.08 0.16 0.45 0.61 0.08 0.15 0.46 0.61
P
720 0.08 0.06 0.03 0.02 0.04 0.03 0.01 0.01

 

Note. N = Sample Size; T= series length; 7’00 = average estimate of population intercept;

5700 = standard deviation of population intercept; 55700 = standard error of population

intercept; P700: rejection rate of hypothesis test on population intercept; f1 0 = average

estimate of population slope; $710= standard deviation of population slope; 55710 =

49

 

 

standard error of population slope; P710: rejection rate of hypothesis test on population
slope; ; T20 = average estimate of X; 5720: standard deviation of X; SE7”: standard

error of X; P720: rejection rate of hypothesis test on X.

Table 7. Model 3 - UGM with X as a Fixed Effect - Random Effect Statistics

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150

T 5 10 40 70 5 10 40 70
—2

00 0.91 1.49 5.30 8.97 0.95 1.54 5.44 9.40
P 2

00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
512 0.21 0.1 1 0.03 0.02 0.21 0.1 1 0.03 0.02
P 2

01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
-2

0'01 0.20 0.00 -O.18 -0.20 0.17 -0.01 -0.19 -0.21
P 2

001 0.14 0.06 0.18 0.21 0.33 0.07 0.63 0.74
—2

0'6 0.46 0.80 2.78 4.79 0.47 0.80 2.80 4.79
PA

12 0.08 0.19 0.46 0.62 0.08 0.16 0.46 0.61

 

 

 

 

 

 

 

 

Note. N = Sample Size; T= series length; 53 = average variance of intercepts; P03 =

rejection rate of hypothesis test on variance of intercepts; 512 = average variance of

slopes; P012 = rejection rate of hypothesis test on variance of slopes; 531 = average
correlation between intercepts and slopes; P 031 = rejection rate of hypothesis test on

correlation between intercepts and slopes; 53 = average within-person variance around
linear trajectories; PA 12 = rejection rate of the hypothesis test on the difference between

model fit of Model 2 and Model 3.

50

Model 4: Unconditional Growth Model with X as a Fixed and Random Effect
The addition of a time-varying predictor to the model as both a fixed and random
effect had no effect on the fixed effect estimates of the intercept and Time. Table 8 shows
that both parameters were still approximately zero in all conditions with approximately
nominal rejection rates, confirming hypotheses 12 and 13. Likewise, the fixed effect of X
stayed approximately zero for all conditions, supporting hypothesis 15. However, by
allowing X to vary, the covariance structure of the model more closely approximates the

covariance structure of the data, resulting in the accurate approximation of the standard

 

error relative to the standard deviation for the fixed effect of X. Therefore, hypothesis 16 E

is supported because the rejection rates of the fixed effect of X are no longer inflated and

 

are now approximately nominal regardless of the number of time points.
By adding an additional random effect to the model, the variance of the system is
divided up further. Therefore, the patterns of the variance components of the random

effects for the intercept and Time were slightly ameliorated (see Table 9). This caused the

. . . 2
vanance component of the random effect for the Intercept to be slightly less than 15 and

 

6
the variance component of the random effect for Time to be slightly greater than .5—1‘-’

supporting hypothesis 14. Despite the mathematical proof that showed that the variance
component of the random effect of X should remain constant at some value (Durlauf &
Phillips, 1988), the variance increased as the number of time points increases. Therefore,

hypothesis 17 is not supported. It is possible that a greater number of time points would

need to be observed before the variance stabilized.

51

 

The significance test for the variance components of the random effect for Time
was significant for every replication, across all eight conditions. However, the
significance tests for the variance components of the random effects for the intercept and
X were dependent on both sample size and number of time points. The rejection rate for
the variance component of the random effect for the intercept was one hundred percent in
every condition except the one with the least information (N = 30; T = 5). Even in this
condition the rejection rate was very high at 99.8%. However, for the variance
component of the random effect for X the rejection rate varied considerably based on both
sample size and number of time points. By increasing either sample size or number of
time points the rejection rate became significant 100% of the time. However, in
conditions of low information (N = 30; T = 5) the rejection rate was as low as 9.4%.
Therefore, hypothesis 18 was only partially supported.

Similarly, the deviance statistic between models 3 and 4 was affected by the
amount of information and followed the same pattern of results as the rejection rate of the
variance component of the random effect for X, thus only partially supporting hypothesis
19. This result is not overly surprising, given the behavior of the rejection rates of the
variance components of the random effects of X. The deviance statistic between these
models is a three degree of freedom test to determine if any of the three new parameters
are significantly different from zero. The two new correlations (between the intercept and

X and between X and Time) had rejection rates slightly above the nominal rate (~0.08) so
the test essentially became equivalent to the 12 test for the parameter estimate of the
variance component for the random effect for X. The results here are worrisome because

they would lead researchers to conclude that a relationship exists between the time-

52

 

 

 

varying predictor and the criterion when in fact, no such relationship exists in the data

generating mechanism. Also, these results are once again somewhat counter intuitive

because as researchers gain greater amounts of information through both sample size and

time points (generally thought of as a good thing), researchers are much more susceptible

to spurious results and inferences.

Table 8. Model 4 - UGM with X as a Fixed and Random Effect - Fixed Effect Statistics

 

 

 

 

 

1"”1

VM". I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150
T 5 10 40 70 5 10 40 70
7’00
(5 ) -0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00
700 (0.20) (0.24) (0.41) (0.52) (0.09) (0.10) (0.18) (0.24
SE
700 0.05 0.05 0.05 0.05 0.05 0.04 0.05 0.06
P700 0.20 0.23 0.40 0.51 0.09 0.11 0.18 0.24
710
(S ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
710 (0.10) (0.07) (0.03) (0.02) (0.04) (0.03) (0.01) (0.01
SE
7m 0.06 0.05 0.04 0.05 0.04 0.06 0.06 0.05
P710 0.09 0.06 0.03 0.02 0.04 0.03 0.01 0.01
720
(S ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
720 (0.09) (0.08) (0.08) (0.08) (0.04) (0.03) (0.03) (0.03)
S E
7’20 0.05 0.07 0.05 0.05 0.06 0.05 0.05 0.04
P720 0.09 0.08 0.07 0.08 0.04 0.04 0.03 0.03

 

 

Note. N = Sample Size; T = series length; 7’00 = average estimate of population intercept;

3700: standard deviation of population intercept; 55700 = standard error of population

intercept; P700: rejection rate of hypothesis test on population intercept; 710 = average

estimate of population slope; 5710: standard deviation of population slope; STE—710 =

standard error of population slope; P710 = rejection rate of hypothesis test on population

53

 

slope; ; 5’20 = average estimate of X; 5720: standard deviation of X; S E 720: standard

error of X; P720= rejection rate of hypothesis test on X.

Table 9. Model 4 - UGM with X as a Fixed and Random Effect - Random Effect

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Statistics

N 30 150

T 5 10 40 70 5 10 40 70

—2

00 0.86 1.36 4.53 7.64 0.91 1.40 4.68 7.98
P 2

0'0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

-2

01 0.20 0.1 1 0.03 0.02 0.21 0.11 0.03 0.02
P 2

01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

—2

02 0.05 0.08 0.14 0.16 0.04 0.08 0.14 0.16
P 2

02 0.09 0.61 1.00 1.00 0.33 1.00 1.00 1.00
-2

001 0.20 0.00 -017 -0.20 0.17 0.00 -0.18 -0.20
P031 0.13 0.06 0.18 0.20 0.32 0.07 0.54 0.69
-2

002 -001 0.00 0.00 0.00 0.01 0.01 -0.01 0.00
P631 0.06 0.09 0.09 0.09 0.06 0.06 0.08 0.09
—2

012 0.02 0.01 -001 -001 -0.01 0.00 0.00 0.00
Pofz 0.06 0.07 0.09 0.08 0.06 0.06 0.08 0.08

—2

05 0.44 0.73 2.40 4.08 0.45 0.73 2.41 4.07
P4

12 0.08 0.52 1.00 1.00 0.22 0.99 1.00 1.00

 

 

 

 

 

 

 

 

 

 

 

Note. N = Sample Size; T = series length; 53 = average variance of intercepts; P03 =

rejection rate of hypothesis test on variance of intercepts; 612 = average variance of

slopes; Poi: = rejection rate of hypothesis test on variance of slopes; 522 = average

54

 

. . . . . —2
variance of X; P 0.22 = rejection rate of hypothes1s test on variance of X; 0' 01 = average
correlation between intercepts and slopes; P 0'61: rejection rate of hypothesis test on

. . —2 . .
correlation between intercepts and slopes; 0' 02 = average correlation between intercepts

and X; P032 = rejection rate of hypothesis test on correlation between intercepts and
X; 01122 = average correlation between slopes and X; 1)sz = rejection rate of hypothesis

test on correlation between slopes and X; 562 = average within-person variance around

linear trajectories; P A 12 = rejection rate of the hypothesis test on the difference between
model fit of Model 3 and Model 4.

Model 5: Unconditional Means Model with X as a Fixed Effect

It is common for researchers to exclude the variable Time from RCM models
when using time-varying covariates as predictors. The addition of a time-varying
predictor did not alter the estimates of the fixed effect or variance component of the
random effect for the intercept. Likewise, the significance tests on these components
behaved identically to those in the unconditional means model. Therefore, hypotheses 20
through 22 were supported. The primary focus of this model is the fixed effect estimate
for the time-varying covariate, X. As has been the case in all of the models, the fixed
effect is accurately estimated to be approximately zero in all experimental conditions,
supporting hypothesis 23 (Table 10). Consistent with the rationale given earlier, the
significance test on the fixed effect of X diverged as the number of time points increased.
This caused the rejection rate to steadily increase as the number of time points increased.
Therefore, hypothesis 24 was supported. Only one parameter was added beyond the
unconditional means model, causing the deviance statistic to become a significance test

on that parameter. Consistent with the single parameter significance test, the deviance

55

 

statistic diverged as the number of time points increased leading the rejection rate to

increase with an increased number of time points, supporting hypothesis 25.

Table 10. Model 5 - Unconditional Means Model with X as a Fixed Effect

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150
T 5 l 10 I 40 l 70 5 I 10 l 40 I 70
Fixed Effect Statistics
7’00
( S ) -0.01 0.01 0.01 0.00 -0.01 0.00 0.01 -0.02
700 (0.28) (0.36) (0.67) (0.89) (0.12) (0.17) (0.31) (0.39)
SE
700 0.06 0.05 0.05 0.05 0.04 0.06 0.05 0.04
P700 0.26 0.35 0.66 0.87 0.12 0.16 0.30 0.40
516
(S ) 0.00 0.00 -001 0.00 0.00 0.00 0.00 0.00
710 (0.12) (0.12) (0.11) (0.11) (0.06) (0.05) (0.05) (0.05
SE71), 0.17 0.32 0.57 0.69 0.19 0.30 0.60 0.73
P710 0.08 0.06 0.03 0.02 0.04 0.03 0.01 0.01
Random Effect Statistics
(73 1.92 3.51 13.06 22.73 1.98 3.64 13.48 23.49
P 2
0'0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
552 0.99 1.81 6.74 11.72 1.00 1.83 6.80 11.83
P4
12 0.18 0.33 0.59 0.71 0.19 0.31 0.60 0.73

 

Note. N = Sample Size; T = series length; 5’00 = average estimate of population intercept;

5700 = standard deviation of population intercept; S E 700 = standard error of population

intercept; P700: rejection rate of hypothesis test on population intercept; 7’10 = average

estimate of X; S = standard deviation of X; SE = standard error of X; P =
7'10 710 710

rejection rate of hypothesis test on X; 0% = average variance of intercepts; P03 =

rejection rate of hypothesis test on variance of intercepts; 53 = average within-person

56

 

 

 

 

1111.3

11.1w

{’1'

ho

variance around linear trajectories; P412 = rejection rate of the hypothesis test on the

difference between model fit of unconditional means model and Model 5.

Model 6: Unconditional Growth Model with X as a Fixed and Random Effect

The final model that was tested is a model where a time-varying predictor, X, is
inserted as both a fixed and random effect and the variable Time is excluded. This model
takes on the functional form of the unconditional growth model. Table 11 shows that the
fixed effects for both the intercept and X are well behaved and estimated to be
approximately zero, supporting hypotheses 26 and 29. Similarly, hypotheses 27 and 30
were supported because the standard errors of those fixed effects closely approximate the
corresponding standard deviations leading the significance tests to remain approximately
nominal.

As seen in Table 12, the variance component of the random effect of the intercept

increased as the number of time points increases. However, the value did not converge to

_ 2T
the function 1—5. Instead, the value was estimated to be greater in every model. Therefore,

hypothesis 28 was only partially supported. The significance test for the variance
component of the random effect of the intercept was significant in every replication of
every condition in support of hypothesis 32. The variance component of the random
effect for X also increased as the number of time points increased. This is contrary to the
expectations laid out from Durlauf and Phillips (1988). This finding is consistent
however, with the results for Model 4 where X is included as a fixed and random effect in
addition to the variable Time. Therefore, hypothesis 3] was not supported. The

significance test for the variance component of the random effect of X was significant the

57

majority of the time. In the conditions with few people and few time points (N = 30; T =

5, 10) the rejection rate were not quite at one hundred percent (72% and 99%

respectively). Therefore, hypothesis 32 was only partially supported.

The deviance statistic between Models 5 and 6 tests whether the addition of the

variance component of the random effect for X and the correlation between X and

intercepts is significant. The correlation is significant only slightly more than the nominal

rate, so the primary statistic being analyzed is the variance component of the random

effect. Not surprisingly then, the rejection rate of the deviance statistic follows the same

pattern as the rejection rate of the variance component of the random effect for X. For

conditions with both few people and few time points (N = 30; T= 5, 10) the rejection rate

is slightly less than one hundred percent (68% and 99% respectively), partially

supporting hypothesis 33.

Table l 1. Model 6 - UMM with X as a Fixed and Random Effect - Fixed Effect Statistics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150

T 5 10 40 70 5 10 40 70
700
(S ) 0.00 0.01 0.01 0.00 -0.01 0.00 0.00 -0.02
700 (0.27) (0.32) (0.60) (0.74) (0.11) (0.15) (0.26) (0.34)
SE

700 0.06 0.05 0.05 0.04 0.04 0.05 0.05 0.05
P

700 0.25 0.31 0.56 0.74 0.11 0.14 0.26 0.33
710
(S ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
710 (0.11) (0.11) (0.11) (0.11) (0.05) (0.05) (0.05) (0.05)
SE

710 0.06 0.05 0.05 0.05 0.04 0.03 0.05 0.06
P

710 0.12 0.11 0.11 0.11 0.05 0.05 0.05 0.05

 

58

 

Note. N = Sample Size; T = series length; 7’00 = average estimate of population intercept;

5700 = standard deviation of population intercept; 55700 = standard error of population
intercept; P700: rejection rate of hypothesis test on population intercept; 7’10 = average

estimate of X; S = standard deviation of X; SE = standard error OfX; P7 =
710 2,10 10

rejection rate of hypothesis test on X.

Table 12. Model 6 - UMM with X as a Fixed and Random Effect - Random Effect
Statistics

 

 

 

 

 

 

 

 

 

 

 

 

N 30 150

T 5 10 40 70 5 10 40 70
—2

00 1.58 2.69 9.38 16.28 1.62 2.77 9.66 16.51
P 2

06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
—2

01 0.20 0.27 0.35 0.36 0.20 0.28 0.35 0.37
P012 0.82 1.40 4.87 8.42 0.82 1.39 4.87 8.34
—2

001 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00
P 2

001 0.10 0.11 0.14 0.10 0.10 0.10 0.10 0.11
—2

of 0.72 0.99 1.00 1.00 1.00 1.00 1.00 1.00
PA

12 0.68 0.99 1.00 1.00 1.00 1.00 1.00 1.00

 

 

 

 

 

 

 

 

Note. N = Sample Size; T = series length; 56'" = average variance of intercepts; P03 =
rejection rate of hypothesis test on variance of intercepts; 512 = average variance of X;
P 0.12 = rejection rate of hypothesis test on variance of X; 0131 = average correlation

between intercepts and X; P 031 = rejection rate of hypothesis test on correlation between

. -2 . . . . . .
intercepts and X; 0' 6 = average Within-person variance around linear trajectories; P412

= rejection rate of the hypothesis test on the difference between model fit of Model 5 and
Model 6.

59

 

GeneraiFindings

Across the six models and eight conditions a number of distinct patterns emerged.
The fixed effects estimates were well behaved and accurately estimated to be
approximately zero in all cases. When a corresponding random effect was in the model,
the fixed effect significance tests were also well behaved and stayed approximately at the
nominal rate. If a corresponding random effect was not in the model, however, the fixed
effects significance tests diverged with increased number of time points, leading to a
serious inflation of Type I error rates and rejection rates well beyond the nominal level
(upwards of 70%).

The variance components of the random effects also followed very distinct
patterns across all the models and conditions. The variance component of the intercept
increased as time points increased for every model. Similarly, the variance component of
the random effect for Time decreased as the number of time points increased for every
model. The variance component of the random effect of X slightly increased as the
number of time points increased, contrary to the rationale derived from the distributional
theory of normal regression parameters. When a large amount of information was present
(i.e. large number of people or time points) the variance components were always
significant. For a few models, when information was low (small N and T), the rejection

rates of the variance components dropped slightly below one hundred percent but were

still much higher than the nominal level (01 = 0.05).

The deviance statistics closely followed the rationale presented earlier. When the
only difference between the models was the addition of a fixed effect, the deviance

statistic became another significance test on that parameter; as the number of time points

60

..H' ‘* It -3" an

 

increased, the rejection rate between the models increased toward one hundred percent.
When the difference between models was multiple parameters resulting from the addition
of random effects, the deviance statistics closely mirrored the rejection rates of the
inserted random effect. The correlations were often small and non-significant so once
again, the deviance statistic essentially became a single parameter test for the variance
component of the random effect. As such, in some cases when low amounts of
information were present (small N or T) the rejection rate dropped slightly below one
hundred percent. When information was high, however, (large N or T) the rejection rates

between models were always one hundred percent.

61

 

 

DISCUSSION

Recommendations

The effects of spurious regression in RCM due to stochastic data are quite
dramatic and could easily lead psychological researchers to make faulty statistical and
scientific inferences. It is still unclear as to the prevalence of stochastic processes that
exist in psychology. As mentioned earlier, random walks and stochastic processes are
quite common in other fields (e.g., economics, computer sciences, and physics).
Therefore, it is likely that similar processes exist in the psychological sciences as well. In
a presentation by Kujanin, Braun, and DeShon (2009) two processes, one cognitive and
one behavioral, were identified as being stochastic. They found that relative team
performance (studied through NBA basketball teams) was found to be indistinguishable
from a random walk with drift. Similarly, they found that cognitive perceptions of
confidence ratings over time followed a random walk. It is imperative that scientists
discover other processes that could be stochastic to avoid making the statistical and
scientific inferences that result from analyzing stochastic data with RCM or other
regression models.

The regression model and generalizations of that model make a number of
assumptions about the dependent variable. One of those assumptions is that all trends
present in the dependent variable are purely deterministic. If the dependent variable
contains stochastic trends then researchers are extremely susceptible to the statistical and
inferential mistakes caused by spurious regression as highlighted by this paper.

Alternatively, if only deterministic trends are present in the dependent variable, then

62

 

 

 

 

regardless of the trends present in the predictors, regression models are well behaved. It is
therefore the recommendation of this thesis that psychologists test the dependent variable
of their longitudinal data for the presence of stochastic trends before applying random
coefficient models or other regression models to the data.

Fortunately, methods have been created to distinguish between trajectories with

purely deterministic trends (e. g., Yt= (X + [31 + Q) and trajectories that are caused at

least in part by stochastic trends (e.g., Yt = Y0+ [31 + 2L1 ft). The most commonly

used test to distinguish between deterministic and stochastic trends is the Augmented

Dickey Fuller (ADF) test (Dickey & Fuller, 1979; Said & Dickey, 1984). For a single

time series, the ADF is

AYt 1' 01 + YYt—i + 2?:1 5pAYt—p+ 6t: (17)

where Yt is the series, AYt = Yt - Yt_1, 01 is the drift, p is the lag order of the
autoregressive process, 0p are the structural autoregressive effects, and E t is the error

term. The null hypothesis associated with this test is, ‘y = 0, says that a series is stochastic

and indistinguishable from a random walk. If the null hypothesis is rejected, then the
series is distinguishable from a random walk and thus deterministic. To run the test, the
analyst needs to determine the lag structure of the time series and if there is a drift. The
ADF is not a high power test and estimating unnecessary parameters for long lags and
drift wastes degrees of freedom. The lag structure of a time series is investigated by
looking at the autocorrelation and partial autocorrelation functions while the existence of

drift in the series is generally assessed visually. The free statistical software R includes

63

 

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the ADF in its set of analytical techniques as well as the autocorrelation and partial
autocorrelation functions.

The standard ADF test is used to evaluate the trends present in a single trajectory.
Psychologists generally gather data on several individuals and a panel version of the ADF
is needed to determine whether the sample of trajectories as a whole is distinguishable
from multiple random walks. To use the test it is necessary to examine the
autocorrelation and partial autocorrelation functions of each series to determine the most
common lag structure across the sample and to determine whether drift exists in at least
the majority of the series. This process is described in most introductory time series texts
(e.g., Enders, 2004). Once these decisions are made, the panel version of the ADF test

developed by Im, Pesaran, and Shin (2003) is computed by applying the standard ADF on

each series and then taking the average value of fin Equation 17. This average value is

compared to a percentile (e. g., 90th or 95th) from the distribution of estimated unit roots
(i.e., y) on random walks for the specified lag order, drift, length, and number of time

series (see Im, Pesaran, & Shin, 2003).

If the null hypothesis of the ADF is not rejected for the dependent variable, then
utilizing RCM or other generalizations of the regression model is appropriate. However,
if the null hypothesis is rejected then researchers are encouraged not to use regression or
any generalizations of the regression model, such as RCM, because the estimates will be
spurious.

A number of alternative models have been developed that can handle data with
the complex properties of stochastic trends. The two most common approaches to dealing

with stochastic data are the use of autoregressive-integrated-moving average (ARIMA)

64

models and structural time series models (Harvey, 1989, 1997). ARIMA models
difference the data until the resulting trends are solely deterministic. Then, they allow
researchers to input variables to predict the deterministic component. This method is
often criticized because by differencing the data many times the resulting deterministic
trend may be nothing like the original data. This leaves the door open for researchers to
make inferential mistakes by determining relationships between types of data that do not
exist in the real world. Structural time series models on the other hand, allow researchers
to insert a stochastic component into the model to filter the stochastic trend out while
simultaneously allowing for predictors of the deterministic trend to be added. This allows
researchers to more accurately model the true underlying process by accounting for the

stochastic trend and then predicting any remaining deterministic trends.

Application to Latent Growth Models

All of the examples and simulations in this thesis were done using random
coefficient modeling. Another common method used to analyze longitudinal data and
answer questions about growth processes is to use a structural equation modeling
approach called latent growth modeling. To the extent that latent growth models mirror
the structure and properties of random coefficient models, they suffer from the same
estimation problems of spurious regression due to the presence of stochastic data. More
specifically, the fixed and random effect estimates are identical and the corresponding
significance tests exhibit the same patterns as documented in RCM. Also, the chi-squared

tests on the latent grth models become exponentially larger with increased number of

65

time points leading to 100% rejection rates for significance tests between nested models
and on individual parameters (e.g., variance components).

The use of latent grth models is often seen as having one large benefit over
RCM in that global fit indices may be used to evaluate overall model fit. Three of the
most common global fit indices that are used are the model chi-squared, the Comparative
Fit Index (CF I), and the Root Mean Squared Error of Approximation (RMSEA) (Hu &
Bentler, 1999). Unfortunately, these fit indices are also biased and behave in systematic
ways when analyzing longitudinal data with stochastic trends. The model chi-squared is
heavily biased by the presence of stochastic data and increases toward infinity with
increased time points, thus always indicating poor model fit. This occurs for the same
reason the chi-squared tests on the variance components for the random effects in RCM
diverged and always indicate significance. Both the CFI and RMSEA are functions of the
model chi-squared and thus exhibit similar distinct patterns as the number of time points

increases. The CF I is given by the equation (Kline, 2005):

A

CF] 1 0g
— '33 9 (18)

where
3M: max(xi4 — dfM)’

Therefore, as the model chi-squared goes toward infinity, so does 3M. As this happens

the numerator in the second part of Equation 18 rapidly dominates the denominator,
leading to the second part of the equation getting large with increased numbers of time

points. This causes the estimate of CFI to go toward zero as the number of time points

66

 

increases, thus indicating poor model fit. Similarly, the equation of RMSEA is (Kline,

2005)

 

3M
dfM(N-1) ’

 

RMSEA = (19)

meaning with increased time points 5M will increase, leading the RMSEA to also

increase, indicating worse model fit. As a result, as the number of time points increase,
the model chi-squared, CF I, and RMSEA will always indicate poor model fit. However,
all of the variance components will indicate a significance amount of variance yet to be
explained, encouraging researchers to search for predictors.

While the fact that the global fit indices indicate poor model fit may be seen as an
advantage, it does not actually alleviate any of the problems caused by analyzing
stochastic data. If researchers attempt to explain additional variance or improve model fit
by adding predictors they will be engaging in the proverbial snipe hunt because no true
predictive relationships exist, and any significant predictors that are found would be due
solely to Type I error. Also, the indication of poor model fit does not address the larger
issue of identifying and understanding the true underlying structure of the data. If the true
data generating mechanism has a stochastic trend, then it should be modeled accordingly
using one of the methods described above. To understand the phenomenon of interest and
make correct statistical and scientific inferences, it is important to test and determine the

structure of the data before deciding on an appropriate analytical tool.

67

Limitations and Future Directions

Despite the shocking nature of the results of this paper, a number of limitations
exist. The Augmented Dickey-Fuller (ADF) test is low power test. Therefore, for the
ADF to properly distinguish between deterministic and stochastic trends, a relatively
large number of time points is needed. If researchers do not have a sufficient number of
time points, the test will always fail to reject the null hypothesis, thus indicating that all
series are stochastic. Similarly, the previously mentioned methods to handle stochastic
data, such as ARIMA, require a large number bf time points to function properly.
Therefore, the recommendations of this paper most directly apply to event/experience
sampling methodologies where a large number of time points is frequently collected.
Unfortunately, since the presence of stochastic trends can impact statistical and scientific
inferences with as few as five time points, all longitudinal studies need to worry about the
results. More research is needed to determine how many time points is sufficient to run
the ADF and to apply ARIMA or structural time series models to longitudinal data. More
research is also needed to determine the exact benefits and limitations of both ARIMA
and structural time series models compared to RCM and other commonly used analytical
techniques in psychology.

Just as more research needs to be done on the proposed data analysis methods,
more research is needed to identify stochastic processes in psychology. One possible
method of discovering stochastic series is by analyzing previously published
event/experience sampling studies and testing the data for the presence of stochastic
trends. Finally, more work also needs to be done to determine how much relative

influence stochastic processes need to have before their presence is problematic.

68

Although more work is needed to determine the exact scope of the problem and what the
best solution is, this paper does provide a good start for recognizing the potential

presence of stochastic trends and the problems that can arise from their presence.

Conclusion

The underlying structure of longitudinal data can have drastic impacts on the
accuracy and quality of statistical estimates and scientific inferences. It is imperative that
researchers determine the nature and structure of their longitudinal data before
determining what analytic method to use. The first step in running any longitudinal data
analysis should be using the Augmented Dickey Fuller (ADF) or some other statistical
test to identify whether stochastic trends are present in the dependent variable. Then, after
determining the underlying structure of the data, an appropriate analytic technique may
be chosen. For data that contain only deterministic trends, random coefficient models or
any other generalization of the regression model is appropriate. However, if stochastic
trends are present in the data, then more complex analytical methods (e.g., ARIMA or
structural time series) must be used to avoid making the statistical and inferential

mistakes that are caused by spurious regression.

69

APPENDIX

70

APPENDIX A

Standard Errors as Approximations of Standard Deviations of the Sampling Distributions
of the Fixed Effects

Durlauf and Phillips (1988) derive the theoretical sampling distribution of
intercepts and slopes for a simple regression (i.e., Y, = a + ,Bt + 6,) applied to data
generated from an underlying random walk process. In the unconditional growth model,

the standard deviations of the sampling distributions for the fixed effects (i.e., 700 and

'y 10) are approximately equal to the standard deviations fi'om Theorem 2.1 of Durlauf and
Phillips (1988) divided by the sample size, N. Thus, the standard deviation of the

intercept (i.e., 700) sampling distribution is:

_2T2

0'0... ——0‘6 ,
15N
(20)

where T is number of time points for each series. The standard deviation of the slope (i.e.,

‘Y10) sampling distribution is:

6
o= —oZ
0 5m 5'

(21)
The standard errors of the fixed effects computed in the unconditional growth
model closely approximate these standard deviations. The standard errors are given by

Raudenbush and Bryk (2002) Equations 3.32 and 3.33

 

SE=JZ§Y=1(W’-AJT1W-1,)‘

(22)

71

where j represents individuals and W]- is a matrix of level 2 predictors for each

‘1 . . .
individual., A]- = S + 052 (Xij) , is the variance-covariance matrix of the random

effects (i.e., intercepts and slopes), X j is the matrix of level 1 predictors, and 052 is the

error variance. The diagonal elements of Equation 22 are needed to compute the standard

errors of the fixed effects. Durlauf and Phillips (1988; Theorem 2.1) provide an

approximation of the diagonal elements (i.e., variance components) of S, and the

.‘J‘l - 11

diagonal elements of the inverse ofXI'Xj are derivable when Time is the only predictor.

Nelson and Kang (1984; Equation 2.9) provide the needed approximation of the error

A..' r uh.

variance.

 

it _

Using the previously mentioned approximations, the standard error on00 is

 

-1

-1
55700 = (g + ((11:71) (4:29)) N ' (23)

 

 

As T increases, the second term in Equation 23 is rapidly dominated by the first term.

Therefore, Equation 23 closely approximates Equation 20.

Using the previously mentioned approximations, the standard error of the slope

fixed effect is,

 

55m: ((31 (1%)(F51—2—rll)-IN)-1° (24)

 

72

As T increases, the second term in Equation 24 is rapidly dominated by the first term.

Therefore, Equation 24 closely approximates Equation 21.

73

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74

 

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