In IQSS/Zelig: Everyone's Statistical Software

Built using Zelig version r packageVersion("Zelig")

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Gamma Regression for Continuous, Positive Dependent Variables with gamma.survey.

Use the gamma regression model if you have a positive-valued dependent variable such as the number of years a parliamentary cabinet endures, or the seconds you can stay airborne while jumping. The gamma distribution assumes that all waiting times are complete by the end of the study (censoring is not allowed).

Syntax

z.out <- zelig(Y ~ X1 + X2, model = "gamma.survey", weights = w, data = mydata)
x.out <- setx(z.out)
s.out <- sim(z.out, x = x.out, x1 = NULL)

Examples

rm(list=ls(pattern="\\.out"))
suppressWarnings(suppressMessages(library(Zelig)))
set.seed(1234)

Example 1: User has Existing Sample Weights

Attach sample data and variable names:

data(api, package="survey")

In this example, we will estimate a model using the percentages of students who receive subsidized lunch and an indicator for whether schooling is year-round to predict California public schools' academic performance index scores:

z.out1 <- zelig(api00 ~ meals + yr.rnd, model = "gamma.survey",
                weights = ~pw, data = apistrat)
summary(z.out1)

Set explanatory variables to their default (mean/mode) values, and set a high (80th percentile) and low (20th percentile) value for "meals," the percentage of students who receive subsidized meals:

x.low <- setx(z.out1, meals= quantile(apistrat$meals, 0.2))
x.high <- setx(z.out1, meals= quantile(apistrat$meals, 0.8))

Generate first differences for the effect of high versus low "meals" on academic performance:

s.out1 <- sim(z.out1, x=x.high, x1=x.low)
summary(s.out1)

Generate a second set of fitted values and a plot:

plot(s.out1)

Example 2: User has Details about Complex Survey Design (but not sample weights)

Suppose that the survey house that provided the dataset excluded probability weights but made other details about the survey design available. We can still estimate a model without probability weights that takes instead variables that identify each the stratum and/or cluster from which each observation was selected and the size of the finite sample from which each observation was selected.

z.out2 <- zelig(api00 ~ meals + yr.rnd, model = "gamma.survey",
                strata=~stype, fpc=~fpc, data = apistrat)
summary(z.out2)

Note that these results are identical to the results obtained when pre-existing sampling weights were used. When sampling weights are omitted, Zelig estimates them automatically for "gamma.survey" models based on the user-defined description of sampling designs. If no description is present, the default assumption is equal probability sampling.

The methods setx()`` andsim()can then be run onz.out2` in the same fashion described in Example 1.

Example 3: User has Replicate Weights

Suppose that the survey house that published these data withheld details about the survey design and instead published replication weights.

For the purpose of illustration, create a set of jk1 replicate weights:

library("survey")
jk1reps <- jk1weights(psu=apistrat$dnum)

Estimate the model regressing api00 on the "meals" "yr.rnd" variables.

z.out3 <- zelig(api00 ~ meals + yr.rnd, model = "gamma.survey",
      data = apistrat, repweights=jk1reps$weights,
      type="JK1")
summary(z.out3)

Set the explanatory variable "meals" at high and low values

x.low <- setx(z.out3, meals= quantile(apistrat$meals, 0.2))
x.high <- setx(z.out3, meals= quantile(apistrat$meals, 0.8))

Generate first differences for the effect of the high versus low concentrations of poverty on school performance

s.out3 <- sim(z.out3, x=x.high, x1=x.low)
summary(s.out3)

Generate a second set of fitted values and a plot:

plot(s.out3)

The user should also refer to the gamma model demo, since gamma.survey models can take many of the same options as gamma models.

Model

• The Gamma distribution with scale parameter $\alpha$ has a stochastic component:

$$ \begin{aligned} Y &\sim& \textrm{Gamma}(y_i \mid \lambda_i, \alpha) \ f(y) &=& \frac{1}{\alpha^{\lambda_i} \, \Gamma \lambda_i} \, y_i^{\lambda_i - 1} \exp -\left{ \frac{y_i}{\alpha} \right}\end{aligned} $$ | for $\alpha, \lambda_i, y_i > 0$.

• The systematic component is given by

$$ \lambda_i = \frac{1}{x_i \beta} $$

Quantities of Interest

• The expected values (qi$ev) are simulations of the mean of the stochastic component given draws of $\alpha$ and $\beta$ from their posteriors:

$$ E(Y) = \alpha \lambda_i. $$

• The predicted values (qi$pr) are draws from the gamma distribution for each given set of parameters $(\alpha, \lambda_i)$.

• If x1 is specified, sim() also returns the differences in the expected values (qi$fd),

$$ E(Y \mid x_1) - E(Y \mid x) $$

.

• In conditional prediction models, the average expected treatment effect (att.ev) for the treatment group is

$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - E[Y_i(t_i=0)] \right}, $$

where $t_i$ is a binary explanatory variable defining the treatment ($t_i=1$) and control ($t_i=0$) groups. Variation in the simulations are due to uncertainty in simulating $E[Y_i(t_i=0)]$, the counterfactual expected value of $Y_i$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $t_i=0$.

• In conditional prediction models, the average predicted treatment effect (att.pr) for the treatment group is

$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - \widehat{Y_i(t_i=0)} \right}, $$

where $t_i$ is a binary explanatory variable defining the treatment ($t_i=1$) and control ($t_i=0$) groups. Variation in the simulations are due to uncertainty in simulating $\widehat{Y_i(t_i=0)}$, the counterfactual predicted value of $Y_i$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $t_i=0$.

Output Values

The Zelig object stores fields containing everything needed to rerun the Zelig output, and all the results and simulations as they are generated. In addition to the summary commands demonstrated above, some simply utility functions (known as getters) provide easy access to the raw fields most commonly of use for further investigation.

In the example above z.out$get_coef() returns the estimated coefficients, z.out$get_vcov() returns the estimated covariance matrix, and z.out$get_predict() provides predicted values for all observations in the dataset from the analysis.

See also

The gammasurvey model is part of the survey package by Thomas Lumley, which in turn depends heavily on glm package. Advanced users may wish to refer to help(svyglm) and help(family).

z5 <- zgammasurvey$new()
z5$references()

IQSS/Zelig documentation built on Dec. 11, 2023, 1:51 a.m.