| micromacro.lm {MicroMacroMultilevel} | R Documentation |
Fitting Micro-Macro Multilevel Linear Models
Description
After computing the adjusted group means of individual-level predictors by adjusted.predictors, use micromacro.lm for estimation results and inferential statistics.
Usage
micromacro.lm(model, adjusted.predictors, y, unequal.groups = NULL)
Arguments
model |
a linear regression model formula, e.g., as.formula(y ~ x1 + x2 ... + xm). |
adjusted.predictors |
a G-by-m data frame, where column variables are group-level predictors and the adjusted group means of individual-level predictors were computed by the |
y |
an array or a G-by-1 numeric matrix that corresponds to the group-level outcome variable in the model. |
unequal.groups |
an optional boolean variable automatically reported by the |
Details
To date, most multilevel methodologies can only unbiasedly model macro-micro multilevel situations, wherein group-level predictors (e.g., city temperature) are used to predict an individual-level outcome variable (e.g., citizen personality). In contrast, this R package enables researchers to model micro-macro situations, wherein individual-level (micro) predictors (and other group-level predictors) are used to predict a group-level (macro) outcome variable in an unbiased way.
To conduct micro-macro multilevel modeling with the current package, one must first compute the adjusted group means with the function adjusted.predictors.
This is because in micro-macro multilevel modeling, it is statistically biased to directly regress the group-level outcome variable on the unadjusted group means of individual-level predictors (Croon & van Veldhoven, 2007).
Instead, one should use the best linear unbiased predictors (BLUP) of the group means (i.e., the adjusted group means), which is conveniently computed by adjusted.predictors.
Once produced by adjusted.predictors, the adjusted group means can be used as one of the inputs of the micromacro.lm function, which reports estimation results and inferential statistics of the micro-macro multilevel model of interest.
If group size is the same across all groups (i.e., unequal.groups = FALSE), then OLS standard errors are reported and used to determine the inferential statistics in this micro-macro model. If group size is different across groups (i.e., unequal.groups = TRUE), however, then the heteroscedasticity-consistent standard errors are reported and used determine the inferential statistics in this micro-macro model (White, 1980).
Value
statistics a summary reports standard inferential statistics on linear regression, e.g., "Estimate", coefficient estimates; "Uncorrected S.E."/"S.E.", OLS standard errors; "Corrected S.E.", heteroskedasticity-consistent standard errors; "df", degree of freedom; "t", Student t statistics; "Pr(>|t|)", two-sided p-value; "r", effect size.
rsquared r squared.
rsquared.adjusted adjusted r squared.
residuals residuals from the model.
fitted.values fitted values from the model.
fstatistic F statistics of the model.
model.formula model formula.
Author(s)
Jackson G. Lu, Elizabeth Page-Gould, & Nancy R. Xu (maintainer, nancyranxu@gmail.com).
References
Akinola, M., Page-Gould, E., Mehta, P. H., & Lu, J. G. (2016). Collective hormonal profiles predict group performance. Proceedings of the National Academy of Sciences, 113 (35), 9774-9779.
Croon, M. A., & van Veldhoven, M. J. (2007). Predicting group-level outcome variables from variables measured at the individual level: A latent variable multilevel model. Psychological Methods, 12(1), 45-57.
White, H. (1980). A heteroskedasticity-consistent covariance estimator and a direct test of heteroskedasticity. Econometrica, 48, 817-838.
See Also
adjusted.predictors for calculating the adjusted group means of the individual-level predictors, and micromacro.summary for a friendly output summary table.
Examples
######## SETUP: DATA GENERATING PROCESSES ########
set.seed(123)
# Step 1. Generate a G-by-q data frame of group-level predictors (e.g., control variables), z.data
# In this example, G = 40, q = 2
group.id = seq(1, 40)
z.var1 = rnorm(40, mean=0, sd=1)
z.var2 = rnorm(40, mean=100, sd=2)
z.data = data.frame(group.id, z.var1, z.var2)
# Step 2. Generate a G-by-p data frame of group-level means for the predictors that will be used to
# generate x.data
# In this example, there are 3 individual-level predictors, thus p = 3
x.var1.means = rnorm(40, mean=50, sd = .05)
x.var2.means = rnorm(40, mean=20, sd = .05)
x.var3.means = rnorm(40, mean=-10, sd = .05)
x.data.means = data.frame(group.id, x.var1.means, x.var2.means, x.var3.means)
# Step 3. Generate two N-by-p data frames of individual-level predictors, x.data
# One of these two data frames assumes unequal-sized groups (Step 3a), whereas the other assumes
# equal-sized groups (Step 3b):
# Step 3a. Generate the individual-level predictors
# In this example, N = 200 and group size is unequal
x.data.unequal = data.frame( group.id=rep(1:40, times=sample( c(4,5,6), 40, replace=TRUE) )[1:200] )
x.data.unequal = merge( x.data.unequal,
data.frame( group.id, x.var1.means, x.var2.means, x.var3.means ), by="group.id" )
x.data.unequal = within( x.data.unequal, {
x.var1 = x.var1.means + rnorm(200, mean=0, sd = 2)
x.var2 = x.var2.means + rnorm(200, mean=0, sd = 6)
x.var3 = x.var3.means + rnorm(200, mean=0, sd = 1.5)
})
# Step 3b. Generate the individual-level predictors
# In this example, N = 200 and group size is equal
x.data.equal = data.frame( group.id=rep(1:40, each=5) )
x.data.equal = merge( x.data.equal, x.data.means, by="group.id" )
x.data.equal = within( x.data.equal, {
x.var1 = x.var1.means + rnorm(200, mean=0, sd = 2)
x.var2 = x.var2.means + rnorm(200, mean=0, sd = 6)
x.var3 = x.var3.means + rnorm(200, mean=0, sd = 1.5)
})
# Step 3. Generate a G-by-1 data frame of group-level outcome variable, y
# In this example, G = 40
y = rnorm(40, mean=6, sd=5)
apply(x.data.equal,2,mean)
# group.id x.var1.means x.var2.means x.var3.means x.var3 x.var2 x.var1
# 20.500000 50.000393 19.994708 -9.999167 -10.031995 20.185361 50.084635
apply(x.data.unequal,2,mean)
# group.id x.var1.means x.var2.means x.var3.means x.var3 x.var2 x.var1
# 20.460000 50.002286 19.994605 -9.997034 -9.983146 19.986111 50.123591
apply(z.data,2,mean)
# z.var1 z.var2
# 0.04518332 99.98656817
mean(y)
# 6.457797
######## EXAMPLE 1. GROUP SIZE IS DIFFERENT ACROSS GROUPS ########
######## Need to use adjusted.predictors() in the same package ###
# Step 4a. Generate a G-by-1 matrix of group ID, z.gid. Then generate an N-by-1 matrix of
# each individual's group ID, x.gid, where the group sizes are different
z.gid = seq(1:40)
x.gid = x.data.unequal$group.id
# Step 5a. Generate the best linear unbiased predictors that are calcualted from
# individual-level data
x.data = x.data.unequal[,c("x.var1","x.var2","x.var3")]
results = adjusted.predictors(x.data, z.data, x.gid, z.gid)
# Note: Given the fixed random seed, the output should be as below
results$unequal.groups
# TRUE
names(results$adjusted.group.means)
# "BLUP.x.var1" "BLUP.x.var2" "BLUP.x.var3" "z.var1" "z.var2" "gid"
head(results$adjusted.group.means)
# BLUP.x.var1 BLUP.x.var2 BLUP.x.var3 group.id z.var1 z.var2 gid
# 1 50.05308 20.83911 -10.700361 1 -0.56047565 98.61059 1
# 2 48.85559 22.97411 -9.957270 2 -0.23017749 99.58417 2
# 3 50.16357 19.50001 -9.645735 3 1.55870831 97.46921 3
# 4 49.61853 21.25962 -10.459398 4 0.07050839 104.33791 4
# 5 50.49673 21.38353 -9.789924 5 0.12928774 102.41592 5
# 6 50.86154 19.15901 -9.245675 6 1.71506499 97.75378 6
# Step 6a. Fit a micro-macro multilevel model when group sizes are different
model.formula = as.formula(y ~ BLUP.x.var1 + BLUP.x.var2 + BLUP.x.var3 + z.var1 + z.var2)
model.output = micromacro.lm(model.formula, results$adjusted.group.means, y, results$unequal.groups)
micromacro.summary(model.output)
# Call:
# micromacro.lm( y ~ BLUP.x.var1 + BLUP.x.var2 + BLUP.x.var3 + z.var1 + z.var2, ...)
#
# Residuals:
# Min 1Q Median 3Q Max
# -13.41505 -2.974074 1.13077 3.566021 6.975819
#
#
# Coefficients:
# Estimate Uncorrected S.E. Corrected S.E. df t Pr(>|t|) r
# (Intercept) 78.1232185 121.5103390 122.1367432 34 0.6396373 0.5266952 0.10904278
# BLUP.x.var1 -0.7589602 1.4954434 1.7177575 34 -0.4418320 0.6614084 0.07555696
# BLUP.x.var2 0.4263309 0.7070773 0.6299759 34 0.6767416 0.5031484 0.11528637
# BLUP.x.var3 0.2658078 2.4662049 2.4051691 34 0.1105152 0.9126506 0.01894980
# z.var1 0.4315941 1.0855707 1.0614535 34 0.4066068 0.6868451 0.06956356
# z.var2 -0.3949955 0.5573789 0.4230256 34 -0.9337390 0.3570228 0.15812040
#
# ---
# Residual standard error: 5.1599 on 34 degrees of freedom
# Multiple R-squared: 0.0400727607, Adjusted R-squared: -0.1010930098
# F-statistic: 0.28387 on 5 and 34 DF, p-value: 0.91869
model.output$statistics
# Estimate Uncorrected S.E. Corrected S.E. df t Pr(>|t|) r
# (Intercept) 78.1232185 121.5103390 122.1367432 34 0.6396373 0.5266952 0.10904278
# BLUP.x.var1 -0.7589602 1.4954434 1.7177575 34 -0.4418320 0.6614084 0.07555696
# BLUP.x.var2 0.4263309 0.7070773 0.6299759 34 0.6767416 0.5031484 0.11528637
# BLUP.x.var3 0.2658078 2.4662049 2.4051691 34 0.1105152 0.9126506 0.01894980
# z.var1 0.4315941 1.0855707 1.0614535 34 0.4066068 0.6868451 0.06956356
# z.var2 -0.3949955 0.5573789 0.4230256 34 -0.9337390 0.3570228 0.15812040
model.output$rsquared
# 0.0400727607
model.output$rsquared.adjusted
# -0.1010930098
######## EXAMPLE 2. GROUP SIZE IS THE SAME ACROSS ALL GROUPS ########
######## Need to use adjusted.predictors() in the same package ######
# Step 4b. Generate a G-by-1 matrix of group ID, z.gid. Then generate an N-by-1 matrix of
# each individual's group ID, x.gid, where group size is the same across all groups
z.gid = seq(1:40)
x.gid = x.data.equal$group.id
# Step 5b. Generate the best linear unbiased predictors that are calcualted from
# individual-level data
x.data = x.data.equal[,c("x.var1","x.var2","x.var3")]
results = adjusted.predictors(x.data, z.data, x.gid, z.gid)
results$unequal.groups
# FALSE
names(results$adjusted.group.means)
# "BLUP.x.var1" "BLUP.x.var2" "BLUP.x.var3" "z.var1" "z.var2" "gid"
results$adjusted.group.means[1:5, ]
# BLUP.x.var1 BLUP.x.var2 BLUP.x.var3 group.id z.var1 z.var2 gid
# 1 50.91373 19.12994 -10.051647 1 -0.56047565 98.61059 1
# 2 50.19068 19.17978 -10.814382 2 -0.23017749 99.58417 2
# 3 50.13390 20.98893 -9.952348 3 1.55870831 97.46921 3
# 4 49.68169 19.60632 -10.612717 4 0.07050839 104.33791 4
# 5 50.28579 22.07469 -10.245505 5 0.12928774 102.41592 5
# Step 6b. Fit a micro-macro multilevel model when group size is the same across groups
model.output2 = micromacro.lm(model.formula, results$adjusted.group.means, y,
results$unequal.groups)
micromacro.summary(model.output2)
# Call:
# micromacro.lm( y ~ BLUP.x.var1 + BLUP.x.var2 + BLUP.x.var3 + z.var1 + z.var2, ...)
#
# Residuals:
# Min 1Q Median 3Q Max
# -12.94409 -1.898937 0.8615494 3.78739 8.444582
#
#
# Coefficients:
# Estimate S.E. df t Pr(>|t|) r
# (Intercept) 135.4109966 134.1478457 34 1.0094161 0.3199052 0.17057636
# BLUP.x.var1 -2.1984308 2.2203278 34 -0.9901379 0.3291012 0.16741080
# BLUP.x.var2 -0.6369600 0.8619558 34 -0.7389706 0.4649961 0.12572678
# BLUP.x.var3 -0.5121002 1.7889594 34 -0.2862559 0.7764192 0.04903343
# z.var1 0.7718147 1.1347170 34 0.6801826 0.5009945 0.11586471
# z.var2 -0.1116209 0.5268130 34 -0.2118795 0.8334661 0.03631307
#
# ---
# Residual standard error: 5.11849 on 34 degrees of freedom
# Multiple R-squared: 0.0554183804, Adjusted R-squared: -0.0834906813
# F-statistic: 0.39895 on 5 and 34 DF, p-value: 0.84607
model.output2$statistics
# Estimate S.E. df t Pr(>|t|) r
# (Intercept) 135.4109966 134.1478457 34 1.0094161 0.3199052 0.17057636
# BLUP.x.var1 -2.1984308 2.2203278 34 -0.9901379 0.3291012 0.16741080
# BLUP.x.var2 -0.6369600 0.8619558 34 -0.7389706 0.4649961 0.12572678
# BLUP.x.var3 -0.5121002 1.7889594 34 -0.2862559 0.7764192 0.04903343
# z.var1 0.7718147 1.1347170 34 0.6801826 0.5009945 0.11586471
# z.var2 -0.1116209 0.5268130 34 -0.2118795 0.8334661 0.03631307
model.output2$rsquared
# 0.0554183804
model.output2$rsquared.adjusted
# -0.0834906813
######## EXAMPLE 3 (after EXAMPLE 2). ADDING A MICRO-MICRO INTERACTION TERM ########
model.formula3 = as.formula(y ~ BLUP.x.var1 * BLUP.x.var2 + BLUP.x.var3 + z.var1 + z.var2)
model.output3 = micromacro.lm(model.formula3, results$adjusted.group.means, y,
results$unequal.groups)
micromacro.summary(model.output3)
# Call:
# micromacro.lm( y ~ BLUP.x.var1 * BLUP.x.var2 + BLUP.x.var3 + z.var1 + z.var2, ...)
#
# Residuals:
# Min 1Q Median 3Q Max
# -13.21948 -2.048324 0.7062639 3.843816 7.924922
#
#
# Coefficients:
# Estimate S.E. df t Pr(>|t|) r
# (Intercept) -1.098875e+03 1962.9182021 33 -0.5598169 0.5793848 0.09699214
# BLUP.x.var1 2.231877e+01 38.9620284 33 0.5728339 0.5706400 0.09922547
# BLUP.x.var2 5.988568e+01 96.0256433 33 0.6236426 0.5371496 0.10792809
# BLUP.x.var3 -9.557605e-01 1.9374178 33 -0.4933167 0.6250560 0.08556050
# z.var1 6.116347e-01 1.1727757 33 0.5215274 0.6054822 0.09041443
# z.var2 -8.556163e-02 0.5331509 33 -0.1604829 0.8734790 0.02792560
# BLUP.x.var1:BLUP.x.var2 -1.209354e+00 1.9186909 33 -0.6303016 0.5328380 0.10906688
#
# ---
# Residual standard error: 5.08795 on 33 degrees of freedom
# Multiple R-squared: 0.0666547309, Adjusted R-squared: -0.103044409
# F-statistic: 0.39278 on 6 and 33 DF, p-value: 0.87831
model.output3$statistics
# Estimate S.E. df t Pr(>|t|) r
# (Intercept) -1.098875e+03 1962.9182021 33 -0.5598169 0.5793848 0.09699214
# BLUP.x.var1 2.231877e+01 38.9620284 33 0.5728339 0.5706400 0.09922547
# BLUP.x.var2 5.988568e+01 96.0256433 33 0.6236426 0.5371496 0.10792809
# BLUP.x.var3 -9.557605e-01 1.9374178 33 -0.4933167 0.6250560 0.08556050
# z.var1 6.116347e-01 1.1727757 33 0.5215274 0.6054822 0.09041443
# z.var2 -8.556163e-02 0.5331509 33 -0.1604829 0.8734790 0.02792560
# BLUP.x.var1:BLUP.x.var2 -1.209354e+00 1.9186909 33 -0.6303016 0.5328380 0.10906688
model.output3$rsquared
# 0.0666547309
model.output3$rsquared.adjusted
# -0.103044409
######## EXAMPLE 4 (after EXAMPLE 2). ADDING A MICRO-MACRO INTERACTION TERM ########
model.formula4 = as.formula(y ~ BLUP.x.var1 + BLUP.x.var2 + BLUP.x.var3 * z.var1 + z.var2)
model.output4 = micromacro.lm(model.formula4, results$adjusted.group.means, y,
results$unequal.groups)
micromacro.summary(model.output4)
# Call:
# micromacro.lm( y ~ BLUP.x.var1 + BLUP.x.var2 + BLUP.x.var3 * z.var1 + z.var2, ...)
#
# Residuals:
# Min 1Q Median 3Q Max
# -12.99937 -1.909645 0.8775397 3.712013 8.46591
#
#
# Coefficients:
# Estimate S.E. df t Pr(>|t|) r
# (Intercept) 129.22731579 146.4817031 33 0.8822079 0.3840456 0.15179313
# BLUP.x.var1 -2.10556192 2.3951160 33 -0.8791064 0.3857003 0.15127172
# BLUP.x.var2 -0.63762927 0.8747645 33 -0.7289153 0.4711953 0.12587857
# BLUP.x.var3 -0.53590189 1.8273917 33 -0.2932605 0.7711594 0.05098372
# z.var1 2.95426548 19.1170600 33 0.1545356 0.8781288 0.02689146
# z.var2 -0.09852267 0.5467583 33 -0.1801942 0.8581021 0.03135236
# BLUP.x.var3:z.var1 0.21489002 1.8788995 33 0.1143702 0.9096374 0.01990534
#
# ---
# Residual standard error: 5.11747 on 33 degrees of freedom
# Multiple R-squared: 0.0557926451, Adjusted R-squared: -0.1158814195
# F-statistic: 0.32499 on 6 and 33 DF, p-value: 0.91909
model.output4$statistics
# Estimate S.E. df t Pr(>|t|) r
# (Intercept) 129.22731579 146.4817031 33 0.8822079 0.3840456 0.15179313
# BLUP.x.var1 -2.10556192 2.3951160 33 -0.8791064 0.3857003 0.15127172
# BLUP.x.var2 -0.63762927 0.8747645 33 -0.7289153 0.4711953 0.12587857
# BLUP.x.var3 -0.53590189 1.8273917 33 -0.2932605 0.7711594 0.05098372
# z.var1 2.95426548 19.1170600 33 0.1545356 0.8781288 0.02689146
# z.var2 -0.09852267 0.5467583 33 -0.1801942 0.8581021 0.03135236
# BLUP.x.var3:z.var1 0.21489002 1.8788995 33 0.1143702 0.9096374 0.01990534
model.output4$rsquared
# 0.0557926451
model.output4$rsquared.adjusted
# -0.1158814195