Mixed Effects Models - Fixed Effects

Mixed Effects Models!
• Next 3 weeks: Basics of a mixed effects
analysis with continuous/numerical variables
• This week: Fixed effects (effects of interest)
• Next 2 weeks: Random effects (e.g., subjects,
classrooms, items, firms, dyads/couples)
• After that: categorical variables
• As predictors
• As outcomes

Week 3.1: Fixed Effects
! Sample Dataset & Research Questions
! Theoretical Model
! lme4 in R
! Parameter Interpretation
! Slopes
! Intercept
! Hypothesis Testing
! t-test
! Confidence Interval
! More About Model Fitting

Introduction to Fixed Effects
• Canvas: Modules " Week 3.1
• Team problem-solving task in organizations
• Assemble a landline phone
• 10 teams at each of 11 companies (total 110)
• Dependent measure: Minutes
taken to complete the task
• Predictor variables:
• Number of newcomers on the
team
• Average years of experience
Lewis, Belliveau, Herndon, & Keller (2007)

• Canvas: Modules " Week 3.1
• Team problem-solving task in organizations
• Assemble a landline phone
• 10 teams at each of 11 companies (total 110)

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
+ Company
Baseline
• Predicting one variable as a function of others

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
+ Company
Baseline
NEXT
WEEK!

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
Fixed effects that we’re
trying to model

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
γ00
(Bryk & Raudenbush, 1992; Quene & van den Bergh, 2004, 2008)
THEORY AND DEFINITION:

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
γ00
x1i(j) x2i(j)
THEORY AND DEFINITION:

• Relationship of number of newcomers to completion
time in minutes is probably not 1:1
Regression line
relating number of
newcomers to
completion time

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
• γ20 = slope of line relating newcomers to time taken
• One of the fixed effects we want to find this out
• How do newcomers affect completion time in this task?
γ00
γ10x1i(j) γ20x2i(j)

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
• Can we determine the exact completion time
based on team composition?
γ00

• Probably not.

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
• Probably not.
• These variables just provide our best guess
• The expected value
γ00
E(Yi(j))

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
• To represent the actual observation, we
need to add an error term
• Discrepancy between expected & actual value
γ00
yi(j)
+ Error

=
Minutes
taken to
assemble
phone
Newcomers
+ +
Years of
Experience
Baseline
• To represent the actual observation, we
need to add an error term
• Discrepancy between expected & actual value
γ00
yi(j)
+ ei(j)
Error

• What if we aren’t interested in predicting
specific values?
• e.g., We want to know whether a variable matters or
the size of its effect
• But: We learn this from asking whether
and how an independent variable predicts
the dependent variable
• If number of newcomers significantly predicts what
the completion time will be, there’s a relation

Running the Model in R
L
M
E
R
inear
ixed
ffects
egression

• Time to fit our first model!
• model1 <-
lmer(
MinutesTaken ~
1 +
Newcomers + YearsExperience
+ (1|CompanyID)
, data=problemsolving)
• Here it is as a single line:
• model1 <- lmer(MinutesTaken ~ 1 + Newcomers
+ YearsExperience + (1|CompanyID),
data=problemsolving)
Name of our model, like naming a dataframe
Linear mixed effects regression (function name)
Dependent measure comes before the ~
Intercept (we’ll discuss this more very soon)
Variables of interest (fixed
effects)
Random effect variable
Name of the dataframe where
your data is

• Time to fit our first model!
• model1 <-
lmer(
MinutesTaken ~
1 +
Newcomers + YearsExperience
+ (1|CompanyID)
, data=problemsolving)
• Quick note: This version of the model makes assumptions
about the random effects that might or might not be true.
We’ll deal with this in the next two weeks when we discuss
random effects.
Name of our model, like naming a dataframe
Linear mixed effects regression (function name)
Dependent measure comes before the ~
Intercept (we’ll discuss this more very soon)
Variables of interest (fixed
effects)
Random effect variable
Name of the dataframe where
your data is

• Where are my results?
• Just like with a dataframe, we’ve saved
them in something we can view later
• To view the model results:
• model1 %>% summary()
• Or whatever your model name is
• Saving the model makes it easy to compare
models later or to view your results again

Sample Model Results
Formula: Variables
you included
Data: Dataframe you
ran this model on
Check that these two
matched what you wanted!
Relevant to model fitting.
Will discuss soon.
Random effects = next week!
Number of observations,
# of clustering groups
Results for fixed effects of
interest
Correlations between effects
• Probably don’t need to
worry about this unless
correlations are very high
(Friedman & Wall, 2005;
Wurm & Fisicaro, 2014)

Parameter Estimates
• Estimates are the γ values from the model
notation
• In our sample:
• Each additional newcomer ≈ another 2.2 minutes of
solving time
• Each additional year of experience ≈ 0.10 decrease in
solving time
• Each of these effects is while holding the others constant
• Core feature of multiple regression!
• Don’t need to do residualization for this
(Wurm & Fisicaro, 2014)

Parameter Estimates
• What about the intercept?
• Let’s go back to our theoretical model
=
Minutes Newcomers
+ +
Years of
Experience
Baseline
γ00
γ10 *x1i(j) γ20 * x2i(j)
E(Yi(j))

Parameter Estimates
=
Minutes Newcomers
+ +
Years of
Experience
Baseline
γ00
2.16 *x1i(j) -0.09 * x2i(j)
E(Yi(j))

Parameter Estimates
• What if a team had 0 newcomers (x1i(j) = 0), and 0
years of prior experience (x2i(j) = 0)?
=
Minutes Newcomers
+ +
Years of
Experience
Baseline
γ00
2.16 * 0 -0.09 * 0
E(Yi(j))

Parameter Estimates
• What if a team had 0 newcomers (x1i(j) = 0), and 0
years of prior experience (x2i(j) = 0)?
• Even in this case, we wouldn’t expect the team to
solve the problem in 0 minutes (that’s ridiculous!)
=
Minutes Newcomers
+ +
Years of
Experience
Baseline
γ00
0 0
E(Yi(j))

Parameter Estimates
• Intercept is what we expect when all other predictor
variables are equal to 0
• Here, a team with 0 newcomers but 0 years prior
experience would be expected to solve the problem in
38.25 minutes
=
Minutes Newcomers
+ +
Years of
Experience
Intercept
38.25 0 0
E(Yi(j))

Parameter Estimates
• Equation for a
line (courtesy
middle school)
• y = mx + b
• m: Slope. How
steep is the
line?
• How
strongly
does IV
affect DV?
Slope = 2.16
For every
newcomer, average
solve time increases
by 2.16 min.

Parameter Estimates
• Equation for a
line (courtesy
middle school)
• y = mx + b
• m: Slope. How
steep is the
line?
• b: Intercept.
Where does it
cross y-axis?
• What DV is
expected
when IV is
0?
38.25

Parameter Estimates
• Default is to include the intercept in the model,
and that’s appropriate
• model1 <- lmer(MinutesTaken ~ 1 + Newcomers +
YearsExperience + (1|CompanyID),
data=problemsolving)
• If we eliminated the intercept, we would be
assuming that the DV is exactly 0 when all
predictors are 0
• That would be a more constrained model—makes more
assumptions about the data
• We allowed our model to freely estimate the intercept, and we
came up with 38.25

Parameter Estimates
WHERE THE
@#$^@$ ARE
MY P-VALUES!?

Hypothesis Testing—t test
• Reminder of why we do inferential statistics
• We know there’s some relationship between
newcomers & problem-solving time in our
sample

• Reminder of why we do inferential statistics
• We know there’s some relationship between
newcomers & problem-solving time in our
sample
• But:
• Would this hold true for all people (the
population) doing group problem-solving?
• Or is this sampling error? (i.e., random chance)

• Newcomr effect in our sample estimated to be
2.16 min. … is this good evidence of an effect
in the population?
• Would want to compare relative to a measure
of sampling error
t =
Estimate
Std. error =
2.1642
0.2690

• We don’t have p-values (yet), but do we
have a t statistic
• Effect divided by its standard error (as with any t
statistic)
• A t test comparing this γ estimate to 0
• 0 is the γ expected under the null hypothesis that
this variable has no effect

Point—Counterpoint
Great! A t value. This will be really helpful for
my inferential statistics.
But you also need the degrees of freedom!
And degrees of freedom are not exactly
defined for mixed effects models. GOT YOU!
But, we can estimate the degrees of freedom.
Curses! Foiled again!

Hypothesis Testing—lmerTest
• Another add-on
package, lmerTest,
that estimates the d.f.
for the t-test
• Similar to correction for
unequal variance
Kuznetsova, Brockhoff, & Christensen, 2017

• Once we have lmerTest installed,
need to load it … remember how?
• library(lmerTest)
• With lmerTest loaded, re-run the
lmer() model, then get its summary
• Will have p-values
• In the future, no need to run model twice.
Can load lmerTest from the beginning
• This was just for demonstration purposes

ESTIMATED degrees of freedom – note that
it’s possible to have non-integer numbers
because it’s an estimate
p-value
• What does 1.84e-12 mean?
• Scientific notation shows up if a number is very
large or very small
• 1.84e-12 = 1.84 x 10-12 = .00000000000184
• This p-value is very small! Significant effect of
number of newcomers on problem solving time

p-value
• What does 1.84e-12 mean?
• Scientific notation shows up if a number is very
large or very small
• 1.84e-12 = 1.84 x 10-12 = .00000000000184
• e-12: Move the decimal 12 places to the left
• Can copy & paste the scientific notation into the R
command line to get the regular number

p-value
• What about the other variables?
• Years of experience: Not significant
• A numerical relation, but not significant effect it
generalizes to the broader population
• Intercept significantly differs from 0
• Even if 0 newcomers and 0 years of experience, still
takes time to solve the problem (duh!)

Confidence Intervals
• 95% confidence intervals are:
• Estimate (1.96 * std. error)
• Try calculating the confidence interval for the
newcomer effect
• This is slightly anticonservative
• In other words, with small samples, CI will be too
small (elevated risk of Type I error)
• But OK with even moderately large samples

Confidence Intervals
• http://www.scottfraundorf.com/statistics.html
• Another add-on package: psycholing
• Includes summaryCI() function that does this
for all fixed effects

More on Model Fitting
• We specified the formula
• How does R know what the right γ
values are for this model?

• Solve for x:
• 2(x + 7) = 18

2(x + 7) = 18
• Two ways you might solve this:
• Use algebra
• 2(x+7) = 18
• x+7 = 9
• x = 2
• Guaranteed to give you the right answer
• Guess and check:
• x = 10? -> 34 = 18 Way off!
• x = 1? -> 9 = 18 Closer!
• x = 2? -> 18 = 18 Got it!
• Might have to check a few numbers
ANALYTIC
SOLUTION
NON-
ANALYTIC
SOLUTION

• Two ways you might solve this:
• t-test: Simple formula you can solve with algebra
• Mixed effects models: Need to search for the best
estimates
ANALYTIC
SOLUTION
NON-
ANALYTIC
SOLUTION

• What, specifically, are we searching for?
• Probability: Chance of an outcome given model
• Given a fair coin, probability of heads is 50%
• Likelihood: Chance of a model given data
• Given 52 heads out of 100, is this likely to be a fair coin?
• Looking for the parameter estimates that result in
a model with the highest likelihood
• Maximum likelihood estimation

• Not guessing randomly. Looks
for better & better parameters
until it converges on the
solution
• Like playing “warmer”/“colder”

Model Fitting—Implications
• More complex models take more time to fit
• model1 <- lmer(MinutesTaken ~ 1 +
Newcomers + YearsExperience +
(1|CompanyID), data=problemsolving,
verbose=2)
• verbose=2 shows R’s steps in the search
• Probably don’t need this; just shows you how it works
• Possible for model to fail to converge on a
set of parameters
• Issue comes up more when you have more
complex models (namely, lots of random effects)
• We’ll talk more in a few weeks about when this
might happen & what to do about it

Mixed Effects Models - Fixed Effects

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