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Influence.ME provides tools for detecting influential data in mixed effects models. The application of these models has become common practice, but the development of diagnostic tools has lagged behind. influence.ME calculates standardized measures of influential data for the point estimates of generalized mixed effects models, such as DFBETAS, Cook's distance, as well as percentile change and a test for changing levels of significance. influence.ME calculates these measures of influence while accounting for the nesting structure of the data. The package and measures of influential data are introduced, a practical example is given, and strategies for dealing with influential data are suggested.
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CONTRIBUTED ARTI CLE 1
influence.ME: Tools for Detecting
Influential Data in Mixed Effects Models
by Rense Nieuwenhuis, Manfred te Grotenhuis, and Ben
Pelzer
Abstract influence.ME provides tools for de-
tecting influential data in mixed effects mod-
els. The application of these models has become
common practice, but the development of diag-
nostic tools has lagged behind. influence.ME
calculates standardized measures of influential
data for the point estimates of generalized mixed
effects models, such as DFBETAS, Cook’s dis-
tance, as well as percentile change and a test for
changing levels of significance. influence.ME
calculates these measures of influence while ac-
counting for the nesting structure of the data.
The package and measures of influential data
are introduced, a practical example is given, and
strategies for dealing with influential data are
suggested.
The application of mixed effects regression models
has become common practice in the field of social sci-
ences. As used in the social sciences, mixed effects re-
gression models take into account that observations
on individual respondents are nested within higher-
level groups such as schools, classrooms, states, and
countries (Snijders and Bosker,1999), and are often
referred to as multilevel regression models. Despite
these models’ increasing popularity, diagnostic tools
to evaluate fitted models lag behind.
We introduce influence.ME (Nieuwenhuis,
Pelzer, and Te Grotenhuis,2012), an R-package
that provides tools for detecting influential cases in
mixed effects regression models estimated with lme4
(Bates and Maechler,2010). It is commonly accepted
that tests for influential data should be performed
on regression models, especially when estimates are
based on a relatively small number of cases. How-
ever, most existing procedures do not account for
the nesting structure of the data. As a result, these
existing procedures fail to detect that higher-level
cases may be influential on estimates of variables
measured at specifically that level.
In this paper, we outline the basic rationale on de-
tecting influential data, describe standardized mea-
sures of influence, provide a practical example of the
analysis of students in 23 schools, and discuss strate-
gies for dealing with influential cases. Testing for
influential cases in mixed effects regression models
is important, because influential data negatively in-
fluence the statistical fit and generalizability of the
model. In social science applications of mixed mod-
els the testing for influential data is especially im-
portant, since these models are frequently based on
large numbers of observations at the individual level
while the number of higher level groups is relatively
small. For instance, Van der Meer, Te Grotenhuis,
and Pelzer (2010) were unable to find any country-
level comparative studies involving more than 54
countries. With such a relatively low number of
countries, a single country can easily be overly in-
fluential on the parameter estimates of one or more
of the country-level variables.
Detecting Influential Data
All cases used to estimate a regression model exert
some level of influence on the regression parameters.
However, if a single case has extremely high or low
scores on the dependent variable relative to its ex-
pected value — given other variables in the model,
one or more of the independent variables, or both
— this case may overly influence the regression pa-
rameters by ’pulling’ the estimated regression line
towards itself. The simple inclusion or exclusion of
such a single case may then lead to substantially dif-
ferent regression estimates. This runs against dis-
tributional assumptions associated with regression
models, and as a result limits the validity and gener-
alizability of regression models in which influential
cases are present.
The analysis of residuals cannot be used for the
detection of influential cases (Crawley,2007). Cases
with high residuals (defined as the difference between
the observed and the predicted scores on the depen-
dent variable) or with high standardized residuals
(defined as the residual divided by the standard de-
viation of the residuals) are indicated as outliers.
However, an influential case is not always an out-
lier. On the contrary: a strongly influential case dom-
inates the regression model in such a way, that the
estimated regression line lies closely to this case. Al-
though influential cases thus have extreme values
on one or more of the variables, they can be onliers
rather than outliers. To account for this, the (standard-
ized) deleted residual is defined as the difference be-
tween the observed score of a case on the dependent
variable, and the predicted score from the regression
model that is based on data from which that case was
removed.
Just as influential cases are not necessarily out-
liers, outliers are not necessarily influential cases.
This also holds for deleted residuals. The reason
for this is that the amount of influence a case exerts
on the regression slope is not only determined by
how well it’s (observed) score is fitted by the spec-
ified regression model, but also by its score(s) on the
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CONTRIBUTED ARTI CLE 2
independent variable(s). The degree to which the
scores of a case on the independent variable(s) are
extreme is indicated by the leverage of this case. A
higher leverage means more extreme scores on the
independent variable(s), and a greater potential of
overly influencing the regression outcomes. How-
ever, if a case has very extreme scores on the inde-
pendent variable(s) but is fitted very well by a regres-
sion model, and if this case has a low deleted (stan-
dardized) residual, this case is not necessarily overly
influencing the outcomes of the regression model.
Since neither outliers, nor cases with a high lever-
age, are necessarily influential, a different procedure
is required for detecting influential cases. The basic
rationale behind measuring influential cases is based
on the principle that when single cases are iteratively
omitted from the data, models based on these data
should not produce substantially different estimates.
If the model parameters change substantially after a
single case is excluded, this case may be regarded
as too influential. However, how much change in
the model parameters is acceptable? To standard-
ize the assessment of how influential a single case
is, several measures of influence are commonly used.
First, DFBETAS is a standardized measure of the ab-
solute difference between the estimate with a partic-
ular case included and the estimate without that par-
ticular case (Belsley, Kuh, and Welsch,1980). Second,
Cook’s distance provides an overall measurement of
the change in all parameter estimates, or a selection
thereof (Cook,1977). In addition, we introduce the
measure of percentile change and a test for changing
levels of significance of the fixed parameters.
Up to this point, this discussion on influential
data was limited to how single cases can overly in-
fluence the point estimates (or BETAS) of a regres-
sion model. Single cases, however, can also bias the
confidence intervals of these estimates. As indicated
above, cases with high leverage can be influential
because of their extreme values on the independent
variables, but not necessarily are. Cases with high
leverage but a low deleted residual compress stan-
dard errors, while cases with low leverage and a high
deleted residual inflate standard errors. Inferences
made to the population from models in which such
cases are present, may be incorrect.
Detecting Influential Data in Mixed Ef-
fects Models
Other options are available in R that help evaluat-
ing the fit of regression models, including the de-
tection of influential data. The base R installation
provides various plots for regression models, includ-
ing but not limited to plots showing residuals versus
the fitted scores, Cook’s distances, and the leverage
versus the deleted residuals. The latter plot can be
used to detect cases that affect the inferential prop-
erties of the model, as discussed above. These plots,
however, are not available for mixed effects models.
The LMERConvenienceFunctions package provides
model criticism plots, including the density of the
model residuals and the fitted values versus the stan-
dardized residuals (Tremblay,2012). However, while
this package works with the lme4, it only is applica-
ble for linear mixed effects models.
The influence.ME package introduced here con-
tributes to these existing options, by providing sev-
eral measures of influential data for generalized mixed
effects models. The limitation is that, unfortunately,
as far as we are aware, the measure of leverage was
not developed for generalized mixed effects mod-
els. Consequently, the current installment of influ-
ence.ME emphasizes detecting the influence of cases
on the point estimates of generalized mixed effect
models. It does, however, provide a basic test for de-
tecting whether single cases change the level of sig-
nificance of an estimate, and therefore the ability to
make inferences from the estimated model.
To apply the logic of detecting influential data to
generalized mixed effects models, one has to mea-
sure the influence of a particular higher level group
on the estimates of a predictor measured at that level.
The straightforward way is to delete all observations
from the data that are nested within a single higher
level group, then re-estimate the regression model,
and finally evaluate the change in the estimated re-
gression parameters. This procedure is then repeated
for each higher-level group separately.
The "influence" function in the influence.ME
package performs this procedure automatically, and
returns an object containing information on the pa-
rameter estimates excluding the influence of each
higher level group separately. The returned object of
class "estex" (ESTimates EXcluding the influence of a
group) can then be passed on to one of the functions
calculating standardized measures of influence (such
as DFBETAS and Cook’s Distance, discussed in more
detail in the next section). Since the procedure of the
"influence" function entails re-estimating mixed
effects models several times, this can be computa-
tionally intensive. Unlike the standard approach in
R, we separated the estimation procedure from cal-
culating the measures of influence themselves. This
allows the user to process a single model once using
the "influence" function, and then to evaluate it us-
ing various measures and plots of influence.
In detecting influential data in mixed effects mod-
els, the key focus is on changes in the estimates of
variables measured at the group-level. However,
most mixed effects regression models estimate the ef-
fects of both lower-level and higher-level variables
simultaneously. Langford and Lewis (1998) devel-
oped a procedure in which the mixed effects model
is modified to neutralize the group’s influence on
the higher-level estimate, while at the same time al-
lowing the lower-level observations nested within
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CONTRIBUTED ARTI CLE 3
that group to help estimate the effects of the lower-
level predictors in the model. For each higher-level
unit evaluated based on this method, the intercept-
vector of the model is set to 0, and an (additional)
dummy variable is added to the model, with score
1 for the respective higher level case. This way,
the case under investigation does not contribute to
the variance of the random intercept, nor to the ef-
fects of variables measured at the group-level. in-
fluence.ME provides this functionality, which is ac-
cessed by specifying delete=FALSE as an option to
the "influence" function. As a result of the specific
modification of the model-specification, this specific
procedure suggested by Langford and Lewis (1998)
does not work when factor-variables are used in the
regression model.
Finally, influence.ME also allows for detecting
the influence of lower-level cases in the mixed effects
model. In social science applications of mixed effects
models, with a great number of lower-level observa-
tions nested in a limited number of groups, this will
not always be feasible. Detecting influence of lower-
level observations is supported for applications in
various disciplines where mixed effects models are
typically applied to only a limited number of obser-
vations per group. This procedure is accessed by
specifying obs=TRUE as an option to the "influence"
function. The "influence" function can either deter-
mine the influence of higher-level cases, or of single
observations, but not both at the same time.
The Outcome Measures
The "influence" function described above returns
an object with information on how much the pa-
rameter estimates in a mixed effects model change,
after the (influence of) observations of higher-level
groups and their individual-level observations were
removed from it iteratively. This returned object can
then be passed on to functions that calculate stan-
dardized measures of influence. influence.ME offers
four such measures, which are detailed in this sec-
tion.
DFBETAS
DFBETAS is a standardized measure that indicates
the level of influence observations have on single
parameter estimates (Fox,2002). Regarding mixed
models, this relates to the influence a higher-level
unit has on the parameter estimate. DFBETAS is cal-
culated as the difference in the magnitude of the pa-
rameter estimate between the model including and
the model excluding the higher level case. This abso-
lute difference is divided by the standard error of the
parameter estimate excluding the higher level unit
under investigation:
DFBETASij =ˆ
γiˆ
γi(j)
se ˆ
γi(j)
in which irefers to the parameter estimate, and jthe
higher-level group, so that ˆ
γirepresents the original
estimate of parameter i, and ˆ
γi(j)represents the es-
timate of parameter i, after the higher-level group j
has been excluded from the data.
In influence.ME, values for DFBETAS in mixed
effects models can be calculated using the func-
tion "dfbetas", which takes the object returned
from "influence" as input. Further options include
parameters to provide a vector of index numbers or
names of the selection of parameters for which DF-
BETAS is to be calculated. The default option of
"dfbetas" is to calculate DFBETAS for estimates of
all fixed effects in the model.
As a rule of thumb, a cut-off value is given for
DFBETAS (Belsley et al.,1980):
2/n
in which n, the number of observations, refers to the
number of groups in the grouping factor under eval-
uation (and not to the number of observations nested
within the group under investigation). Values ex-
ceeding this cut-off value are regarded as overly in-
fluencing the regression outcomes for that specific es-
timate.
Cook’s Distance
Since DFBETAS provides a value for each parame-
ter and for each higher-level unit that is evaluated,
this often results in quite a large number of val-
ues to evaluate (Fox,2002). An alternative is pro-
vided by Cook’s distance, a commonly used mea-
sure of influence. Cook’s distance provides a sum-
mary measure for the influence a higher level unit
exerts on all parameter estimates simultaneously, or
a selection thereof. A formula for Cook’s Distance
is provided (Snijders and Bosker,1999;Snijders and
Berkhof,2008):
C0F
j=1
r+1ˆ
γˆ
γ(j)0ˆ
1
Fˆ
γˆ
γ(j)
in which ˆ
γrepresents the vector of original param-
eter estimates, ˆ
γ(j)the parameter estimates of the
model excluding higher-level unit j, and ˆ
Frepre-
sents the covariance matrix. In influence.ME, the
covariance matrix of the model excluding the higher-
level unit under investigation jis used. Finally, ris
the number of parameters that are evaluated, exclud-
ing the intercept vector.
As a rule of thumb, cases are regarded as too in-
fluential if the associated value for Cook’s Distance
exceeds the cut-off value of (Van der Meer et al.,
2010):
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4
n
in which nto the number of groups in the grouping
factor under evaluation.
In influence.ME, values for Cook’s distance in
mixed effects models are calculated using the func-
tion "cooks.distance", which takes the object re-
turned from "influence" as input. Further op-
tions include parameters to provide a vector of in-
dex numbers or names of the parameters for which
Cook’s Distance is to be calculated. In addition, the
user can specify sort=TRUE to have the values for
Cook’s distance returned in descending order.
As a final note, it is pointed out that if Cook’s dis-
tance is calculated based on a single parameter, the
Cook’s distance equals the squared value of DFBE-
TAS for that parameter. This is also reflected in their
respective cut-off values:
r4
n=2
n
Percentile Change
Depending upon the goal for which the mixed model
is estimated (prediction vs. hypothesis testing), the
use of formal measures of influence as DFBETAS and
Cook’s distance may be less desirable. The reason
for this is that based on these measures it is not im-
mediately clear to what extent parameter estimates
change. For substantive interpretation of the model
outcomes, the relative degree to which a parameter
estimate changes may provide more meaningful in-
formation. A simple alternative is therefore offered
by the function "pchange", which takes the same
input-options as the "dfbetas" function. For each
higher-level group, the percentage of change is cal-
culated as the absolute difference between the pa-
rameter estimate both including and excluding the
higher-level unit, divided by the parameter estimate
of the complete model and multiplied by 100%. A
percentage of change is returned for each parameter
separately, for each of the higher-level units under
investigation. In the form of a formula:
ˆ
γˆ
γ(j)1
ˆ
γ100%
No cut-off value is provided, for determining
what percent change in parameter estimate is con-
sidered too large will primarily depend on the goal
for which the model was estimated and, more specif-
ically, the nature of the hypotheses that are tested.
Test for changes in significance
As discussed above, even when cases are not influen-
tial on the point estimates (BETAS) of the regression
model, cases can still influence the standard errors of
these estimates. Although influence.ME cannot pro-
vide the leverage measure to detect this, it provides
a test for changes in the statistical significance of the
fixed parameters in the mixed effects model.
The "sigtest" function tests whether excluding
the influence of a single case changes the statistical
significance of any of the variables in the model. This
test of significance is based on the test statistic pro-
vided by the lme4 package. The nature of this statis-
tic varies between different distributional families in
the generalized mixed effects models. For instance,
the t-statistic is related to a normal distribution while
the z-statistic is related to binomial distributions.
For each of the cases that are evaluated, the test
statistic of each variable is compared to a test-value
specified by the user. For the purpose of this test, the
parameter is regarded as statistically significant if the
test statistic of the model exceeds the specified value.
The "sigtest" function reports for each variable the
estimated test statistic after deletion of each evalu-
ated case, whether or not this updated test statistic
results in statistical significance based on the user-
specified value, and whether or not this new statis-
tical significance differs from the significance in the
original model. So, in other words, if a parameter
was statistically significant in the original model, but
is no longer significant after the deletion of a specific
case from the model, this is indicated by the out-
put of the "sigtest" function. It is also indicated
when an estimate was not significant originally, but
reached statistical significance after deletion of a spe-
cific case.
Plots
All four measures of influence discussed above, can
also be plotted. The "plot" function takes the output
of the "influence" function to create a dotplot of a
selected measure of influence (cf. Sarkar,2008). The
user can specify which measure of influence is to be
plotted using the which= option. The which= option
defaults to dfbetas. Other options are to select cook
to plot the cook’s distances, pchange to plot the per-
centage change, and sigtest to plot the test statis-
tic of a parameter estimate after deletion of specific
cases.
All plots allow the output to be sorted (by spec-
ifying sort=TRUE and the variable to sort on using
to.sort= (not required for plotting cook’s distances).
In addition, a cut-off value can be specified using
(cutoff=). Values that exceed this cut-off value will
be plotted visually differently, to facilitate the identi-
fication of influential cases. By default, the results for
all cases and all variables are plotted, but a selection
of these can be made by specifying parameters= and
/ or groups=. Finally, by specifying abs=TRUE the ab-
solute values of the measure of influence are plotted.
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Example: students in 23 schools
In our example, we are interested in the relationship
between the degree of structure that schools attempt
to enforce in their classrooms and students’ perfor-
mance on a math test. Could it be that a highly
structured class affects their performance?
The influence.ME package contains the school23
data.frame, that provides information on the per-
formance of 519 students in 23 schools. Measure-
ments include individual students’ score on a math
test, school-level measurements of class structure,
and several additional independent variables. Stu-
dent’s class and school are equivalent in this data,
since only one class per school is available. These
data are a subset of the NELS-88 data (National
Education Longitudinal Study of 1988). The data
are publicly available from the internet: http:
//www.ats.ucla.edu/stat/examples/imm/, and are
reproduced with kind permission of Ita Kreft and Jan
de Leeuw (1998).
First, using the lme4 package, we estimate a mul-
tivariate mixed effects model with students nested in
schools, a random intercept, a measurement of indi-
vidual students’ time spent on math homework, and
a measurement of class structure at the school level.
For the purpose of our example, we assume here that
the math, homework, and structure variables were
correctly measured at the interval level.
library(influence.ME)
data(school23)
school23 <- within(school23,
homework <- unclass(homework))
m23 <- lmer(math ~ homework + structure
+ (1 | school.ID),
data=school23)
print(m23, cor=FALSE)
This results in the summary of the model based
on 23 schools (assigned to object m23), as shown be-
low.
Linear mixed model fit by REML
Formula: math ~ homework +
structure + (1 | school.ID)
Data: school23
AIC BIC logLik deviance REMLdev
3734 3756 -1862 3728 3724
Random effects:
Groups Name Variance Std.Dev.
school.ID (Intercept) 19.543 4.4208
Residual 71.311 8.4446
Number of obs: 519, groups: school.ID, 23
Fixed effects:
Estimate Std. Error t value
(Intercept) 52.2356 5.3940 9.684
homework 2.3938 0.2771 8.640
structure -2.0950 1.3237 -1.583
Based on these estimates, we may conclude that
students spending more time on their math home-
work score better on a math test. Regarding class
structure, however, we do not find a statistically sig-
nificant association with math test scores. But, can
we now validly conclude that class structure does
not influence students’ math performance, based on
the outcomes of this model?
Visual Examination
Since the analysis in the previous section has been
based on the limited number of 23 schools, it is, of
course, possible that observations on single schools
have overly influenced these findings. Before using
the tools provided in the influence.ME package to
formally evaluate this, a visual examination of the re-
lationship between class structure and math test per-
formance, aggregated to the school level, will be per-
formed.
struct <- unique(subset(school23,
select=c(school.ID, structure)))
struct$mathAvg <- with(school23,
tapply(math, school.ID, mean))
dotplot(mathAvg ~ factor(structure),
struct,
type=c("p","a"),
xlab="Class structure level",
ylab="Average Math Test Score")
Class structure level
Average Math Test Score
40
45
50
55
60
2345
Figure 1: Visual examination of class structure and
math performance
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CONTRIBUTED ARTI CLE 6
In the syntax above, a bivariate plot of the ag-
gregated math scores and class structure is created,
which is shown in Figure 1. In this plot, it is clear that
one single school represented in the lower-left corner
of the graph seems to be an outlier, and - more im-
portantly - the non-linear curve shown in this graph
clearly indicates this single school with class struc-
ture level of 2 may overly influence a linear regres-
sion line estimated based on these data.
Calculating measures of influence
In the previous section, based on Figure 1we sus-
pected that the combination in one specific school of
the low average math test results of students, and
the low level of class structure in that school, may
have overly influenced the original analysis of our
data. However, this only is a bivariate examination
of the data, and therefore does not take into account
other variables in the model. Hence, in our exam-
ple, our preliminary conclusion that this may be an
influential case is not controlled for possible effects
of the homework variable. A better test is provided
by standardized measures of influence, as calculated
from the regression model rather than from the raw
data.
The first step in detecting influential data is to de-
termine the extent to which the parameter estimates
in model m23 change, when iteratively each of the
schools is deleted from the data. This is done with
the "influence" function:
estex.m23 <- influence(m23, "school.ID")
The "influence" function takes a mixed effects
regression model as input (here: m23), and the group-
ing factor needs to be specified, which in our case is
school.ID. We assign the output of the ”influence”
function to an object named estex.m23. Below, we
use this object as input to the "dfbetas" function, to
calculate DFBETAS.
dfbetas(estex.m23,
parameters=c(2,3))
This results in a substantial amount of output, a
portion of which is shown below. Only the DFBE-
TAS for the homework and structure variables were
returned, since parameters=c(2,3) was specified.
homework structure
6053 -0.13353732 -0.168139487
6327 -0.44770666 0.020481057
6467 0.21090081 0.015320965
7194 -0.44641247 0.036756281
7472 -0.55836772 1.254990963
...
72292 0.62278508 0.003905031
72991 0.52021424 0.021630219
The numerical output given above by the
"dfbetas" function provides a detailed report of the
values of DFBETAS in the model. For each variable,
as well as for each nesting group (in this example:
each school), a value for DFBETAS is computed and
reported upon. The cut-off value of DFBETAS equals
2/n(Belsley et al.,1980), which in this case equals
2/23 =.41. The estimate for class structure in this
model seems to be influenced most strongly by ob-
servations in school number 7472. The DFBETAS
for the structure variable clearly exceeds the cut-off
value of .41. Also, the estimates of the homework vari-
able changes substantially with the deletion of sev-
eral schools, as indicated by the high values of DF-
BETAS.
A plot (shown in Figure 2) of the DFBETAS is cre-
ated using:
plot(estex.m23,
which="dfbetas",
parameters=c(2,3),
xlab="DFbetaS",
ylab="School ID")
Based on Figure 2, it is clear that both the
structure and the homework variables are highly
susceptible to the influence of single schools. For
the structure variable this is not all that surpris-
ing, since class structure was measured at the school
level and shown in Figure 1to be very likely to be
influenced by a single case: school number 7472.
The observation that high values of DFBETAS were
found for the homework variable, suggests that sub-
stantial differences between these schools exist in
terms of how much time students spend on aver-
age on their homework. Therefore, we suggest that
in mixed effects regression models, both the esti-
mates of individual-level and group-level variables
are evaluated for influential data.
The measure of Cook’s distance allows to deter-
mine the influence a single higher-level group has on
the estimates of multiple variables simultaneously.
So, since the "cooks.distance" function allows to
specify a selection of variables on which the values
for Cook’s distance are to be calculated, this can be
used to limit the evaluation to the measurements at
the group-level exclusively. Note, that whereas DF-
BETAS always relates to single variables, Cook’s dis-
tance is a summary measure of changes on all param-
eter estimates it is based on. Reports on Cook’s dis-
tance thus should always specify on which variables
these values are based.
To continue our example, we illustrate the
"cooks.distance" function on a single variable,
since class structure is the only variable measured at
the school-level. In the example below, we use the
same object that was returned from the "influence"
function. The specification of this function is simi-
lar to "dfbetas", and to create a plot of the cook’s
distances we again use the "plot" function with the
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DFbetaS
School ID
6053
6327
6467
7194
7472
7474
7801
7829
7930
24371
24725
25456
25642
26537
46417
47583
54344
62821
68448
68493
72080
72292
72991
−1.0 −0.5 0.0 0.5 1.0
homework
−1.0 −0.5 0.0 0.5 1.0
structure
Figure 2: DFBETAS of class structure and homework
specification which="cook". We specify two addi-
tional arguments to augment the figure. First, we
specify sort=TRUE to have the resulting Cook’s dis-
tances sorted in a descending order in the figure. The
appropriate cut-off value for Cook’s distance with
23 nesting groups equals to 4/23 =.17. By speci-
fying the cut-off value with cutoff=.17, Cook’s dis-
tances exceeding the specified value are easily identi-
fied in the resulting figure. Thus, to receive both nu-
meric output and a graphical representation (Figure
3), the following specification of "cooks.distance"
and "plot" is given:
cooks.distance(estex.m23,
parameter=3, sort=TRUE)
plot(estex.m23, which="cook",
cutoff=.17, sort=TRUE,
xlab="Cook´s Distance",
ylab="School ID")
The output below shows one value of Cook’s dis-
tance for each nesting group, in this case for each
school.
[,1]
24371 6.825871e-06
72292 1.524927e-05
...
54344 2.256612e-01
7829 3.081222e-01
7472 1.575002e+00
Cook's Distance
School ID
47583
25642
6053
6467
68493
26537
72080
46417
25456
24371
7194
7801
68448
72991
6327
54344
72292
7930
24725
7474
7829
62821
7472
0.0 0.2 0.4 0.6
Figure 3: Cook’s Distance based on class structure
The R Journal Vol. X/Y, Month, Year ISSN 2073-4859
CONTRIBUTED ARTI CLE 8
Only a selection of the output is shown here. A
few schools exceed the cut-off value (in Figure 3
these are indicated with red triangles), but one school
stands out: 7472. Clearly, this school strongly in-
fluences the outcomes regarding the structure vari-
able, as we already suspected based on our bivariate
visual examination in Figure 1.
Testing for Changes in Statistical Signifi-
cance (sigtest)
In the example below, the "sigtest" function is used
to test for changing levels of significance after dele-
tion of each of the 23 schools from our example
model. We are specifically interested in the level
of significance of the structure variable, for which
it was already established above that school with
number 7472 is very influential. Since we observed
a negative effect in the original model, we specify
test=-1.96 to test for significance at a commonly
used value (-1.96) of the test statistic. Note that since
we estimated a normally distributed model, the test
statistic here is the t-value.
sigtest(estex.m23, test=-1.96)$structure[1:10,]
In the example above, we only request the results
for the structure variable and for the first 10 schools.
In the results presented below, three columns are
shown. The first column (Altered.Teststat) shows
the value of the test statistic (here for the structure
variable) after the deletion of the respective schools
(indicated in the row labels). Especially school num-
ber 7472 stands out. In the original model, the test
statistic for the structure variable was -1.583, which
was not significant. When the influence of school
number 7472 is excluded from the model, the test
statistic now is -2.72, which exceeds the selected
value of -1.96 selected by us. That the structure vari-
able would be significant by deletion of school 7472
is indicated in the second column (Altered.Sig). The
Changed.Sig column finally confirms whether the
level of significance of the structure variable (which
was not significant in the original model) changed to
significant after each of the schools was deleted.
In the case of our example, the results for Cook’s
Distance and the results of this test for changing lev-
els of significance both indicate that school number
7472 overly influences the regression outcomes re-
garding the school-level structure variable. Refer-
ring to the discussion on influential data above, how-
ever, we emphasize that this is not necessarily always
the case. Cases can influence the point estimates
without affecting their level of significance, or affect
the level of significance without overly affecting the
point estimate itself. Therefore, both tests should be
performed.
Altered.Teststat Altered.Sig Changed.Sig
6053 -1.326409 FALSE FALSE
6327 -1.688663 FALSE FALSE
6467 -1.589960 FALSE FALSE
7194 -1.512686 FALSE FALSE
7472 -2.715805 TRUE TRUE
7474 -1.895138 FALSE FALSE
7801 -1.534023 FALSE FALSE
7829 -1.045866 FALSE FALSE
7930 -1.566117 FALSE FALSE
24371 -1.546838 FALSE FALSE
Before, using DFBETAS, we identified several
several schools that overly influence the estimate of
the homework variable. We have there performed
sigtest test to evaluate whether deletion of any of
the schools changes the level of significance of the
homework variable. These results are not shown here,
but indicated that the deletion of none of the schools
changed the level of significance of the homework
variable.
Measuring the influence of lower-level ob-
servations
Finally, it is possible that a single lower-level obser-
vation affects the results of the mixed effects model,
especially for data with a limited number of lower-
level observations per group. In our example, this
would refer to a single student affecting the estimates
of either the individual-level variables, the school-
level variables, or both. Here, we test whether one
or more individual students affect the estimate of the
school-level structure variable.
To perform this test, the "influence" function is
used, and obs=TRUE is specified to indicate that single
observations (rather than groups) should be evalu-
ated. The user is warned that this procedure often
will be computationally intensive when the number
of lower-level observations is large.
Next, we request Cook’s Distances specifically for
the structure variable. Since the number of student-
level observations in this model is 519, and cut-off
value for Cook’s Distance is defined as 4/n, the cut-
off value is 4/519 =.0077. The resulting output is
extensive, since a Cook’s Distance is calculated for
any of the 519 students. Therefore, in the example
below, we directly test which of the resulting Cook’s
Distances exceeds the cut-off value.
estex.obs <- influence(m23, obs=TRUE)
cks.d <- cooks.distance(estex.obs, parameter=3)
which(cks.d > 4/519)
The output is not shown here, but the reader can
verify that students with numbers 88 and 89 exert too
much influence on the estimate of the structure vari-
able. Using the sigtest function, however, showed
that the deletion of none of the students from the
The R Journal Vol. X/Y, Month, Year ISSN 2073-4859
CONTRIBUTED ARTI CLE 9
data affected the level of significance of the struc-
ture variable, nor of any of the other variables in the
model.
Dealing with Influential Data
Now that overly influential cases have been identi-
fied in our model, we have to decide how to deal
with them. Generally, there are several strategies,
including getting more data, checking data consis-
tency, adapting model specification, deleting the in-
fluential cases from the model, and obtaining addi-
tional measurements on existing cases to account for
the overly influential cases (Van der Meer et al.,2010;
Harrell, Jr.,2001).
Since overly influential data are a problem es-
pecially encountered in models based on a limited
number of cases, a straightforward remedy would
be to observe more cases in the population of inter-
est. In our example, if we would be able to sample
more schools, it may very well turn out that we ob-
serve several additional schools with a low score on
the structure variable, so that school number 7472 is
no longer influential. Secondly, there may have been
measurement, coding, or transcription errors in the
data, that have lead to extreme scores on one or more
of the variables (i.e. it may be worthwhile, if possible,
to check whether class structure and / or students’
math performance in school 7472 really is that low).
Thirdly, the model specification may be improved. If
the data are used to estimate too complex models, or
if parameterization is incorrect, influential cases are
more likely to occur. Perhaps the structure variable
should have been treated as categorical.
These are all general strategies, but cannot always
be applied. Depending on the research setting, it is
not always feasible to obtain more observations, to
return to the raw data to check consistency, or to re-
duce model complexity or change parameterization.
The fourth strategy, deleting influential cases
from the model, can often be applied. In general,
we suggest deleting influential cases one at the time
and then to re-evaluating the model. Deleting one
or more influential cases from a mixed effects model
is done with the "exclude.influence" function. The
input of this function is a mixed effects model object,
and it returns an updated mixed effects model from
which a specified group was deleted. To illustrate,
we delete school number 7472 (which was identified
as being overly influential) and its individual-level
observations, using the example code below:
m22 <- exclude.influence(m23,
"school.ID", "7472")
print(m22, cor=FALSE)
The "exclude.influence" function takes a mixed
effects model as input, and requires the specification
of the grouping factor (school.ID) and the group to
be deleted (7472). It returns a re-estimated mixed
effects model, that we assign to the object m22. The
summary of that model is shown below:
Linear mixed model fit by REML
Formula: math ~ homework + structure
+ (1 | school.ID)
Data: ..1
AIC BIC logLik deviance REMLdev
3560 3581 -1775 3554 3550
Random effects:
Groups Name Variance Std.Dev.
school.ID (Intercept) 15.333 3.9157
Residual 70.672 8.4067
Number of obs: 496, groups: school.ID, 22
Fixed effects:
Estimate Std. Error t value
(Intercept) 59.4146 5.9547 9.978
homework 2.5499 0.2796 9.121
structure -3.8949 1.4342 -2.716
Two things stand out when this model summary
is compared to our original analysis. First, the num-
ber of observations is lower (496 versus 519), as well
as the number of groups (22 versus 23). More impor-
tantly, though, the negative effect of the structure
variable now is statistically significant, whereas it
was not in the original model. So, now these model
outcomes indicate that higher levels of class structure
indeed are associated with lower math test scores,
even when controlled for the students’ homework
efforts.
Further analyses should repeat the analysis for
influential data, for other schools may turn out to be
overly influential as well. These repetitive steps are
not presented here, but as it turned out, three other
schools were overly influential. However, the sub-
stantive conclusions drawn based on model m22 did
not change after their deletion.
Finally, we suggest an approach for dealing with
influential data, based on Lieberman (2005). He ar-
gues that the presence of outliers may indicate that
one or more important variables were omitted from
the model. Adding additional variables to the model
may then account for the outliers, and improve the
model fit. We discussed above that an influential case
is not necessarily an outlier in a regression model.
Nevertheless, if additional variables in the model
can account for the fact that an observation has ex-
treme scores on one or more variables, the case may
no longer be an influential one.
Thus, adding important variables to the model
may solve the problem of influential data. When the
The R Journal Vol. X/Y, Month, Year ISSN 2073-4859
CONTRIBUTED ARTI CLE 10
observations in a regression model are, for instance,
randomly sampled respondents in a large-scale sur-
vey, it often is impossible to return to these respon-
dents for additional measurements. However, in so-
cial science applications of mixed effects models, the
higher-level groups are often readily accessible cases
such as schools and countries. It may very well be
possible to obtain additional measurements on these
schools or countries, and use these to remedy the
presence of influential data.
Summary
influence.ME provides tools for detecting influen-
tial data in mixed effects models. The application of
these models has become common practice, but the
development of diagnostic tools lag behind. influ-
ence.ME calculates standardized measures of influ-
ential data such as DFBETAS and Cook’s distance,
as well as percentile change and a test for chang-
ing in statistical significance of fixed parameter esti-
mates. The package and measures of influential data
were introduced, a practical example was given, and
strategies for dealing with influential data were sug-
gested.
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Rense Nieuwenhuis
Institute for Innovation and Governance Studies (IGS),
University of Twente
P.O. Box 217, 7500 AE, Enschede
The Netherlands
r.nieuwenhuis@utwente.nl
Manfred te Grotenhuis
Radboud University
Nijmegen
The Netherlands
m.tegrotenhuis@maw.ru.nl
Ben Pelzer
Radboud University
Nijmegen
The Netherlands
b.pelzer@maw.ru.nl
The R Journal Vol. X/Y, Month, Year ISSN 2073-4859
... All analyses were conducted in R 3.02 (cran.org). Final models were evaluated using residual plots, and checked for influential cases using the "influence.ME" package (Nieuwenhuis et al., 2012). ...
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Introduction and Historical PerspectiveTechnical Background Experimental ExperienceSummary Interpretation, and Examples of Diagnosing Actual Data for CollinearityAppendix 3A: The Condition Number and InvertibilityAppendix 3B: Parameterization and ScalingAppendix 3C: The Weakness of Correlation Measures in Providing Diagnostic InformationAppendix 3D: The Harm Caused by Collinearity
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