generalized linear models

Generalized Linear Models (GLMs) in R, Part 4: Options, Link Functions, and Interpretation

August 20th, 2015 by

Last year I wrote several articles (GLM in R 1, GLM in R 2, GLM in R 3) that provided an introduction to Generalized Linear Models (GLMs) in R.

As a reminder, Generalized Linear Models are an extension of linear regression models that allow the dependent variable to be non-normal.

In our example for this week we fit a GLM to a set of education-related data.

Let’s read in a data set from an experiment consisting of numeracy test scores (numeracy), scores on an anxiety test (anxiety), and a binary outcome variable (success) that records whether or not the students eventually succeeded in gaining admission to a prestigious university through an admissions test.

We will use the glm() command to run a logistic regression, regressing success on the numeracy and anxiety scores.


A <- structure(list(numeracy = c(6.6, 7.1, 7.3, 7.5, 7.9, 7.9, 8, 
8.2, 8.3, 8.3, 8.4, 8.4, 8.6, 8.7, 8.8, 8.8, 9.1, 9.1, 9.1, 9.3, 
9.5, 9.8, 10.1, 10.5, 10.6, 10.6, 10.6, 10.7, 10.8, 11, 11.1, 
11.2, 11.3, 12, 12.3, 12.4, 12.8, 12.8, 12.9, 13.4, 13.5, 13.6, 
13.8, 14.2, 14.3, 14.5, 14.6, 15, 15.1, 15.7), anxiety = c(13.8, 
14.6, 17.4, 14.9, 13.4, 13.5, 13.8, 16.6, 13.5, 15.7, 13.6, 14, 
16.1, 10.5, 16.9, 17.4, 13.9, 15.8, 16.4, 14.7, 15, 13.3, 10.9, 
12.4, 12.9, 16.6, 16.9, 15.4, 13.1, 17.3, 13.1, 14, 17.7, 10.6, 
14.7, 10.1, 11.6, 14.2, 12.1, 13.9, 11.4, 15.1, 13, 11.3, 11.4, 
10.4, 14.4, 11, 14, 13.4), success = c(0L, 0L, 0L, 1L, 0L, 1L, 
0L, 0L, 1L, 0L, 1L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 
1L, 1L, 1L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L)), .Names = c("numeracy", 
"anxiety", "success"), row.names = c(NA, -50L), class = "data.frame")
attach(A)
names(A)
[1] "numeracy" "anxiety"  "success"
head(A)
    numeracy anxiety  success
1      6.6    13.8       0
2      7.1    14.6       0
3      7.3    17.4       0
4      7.5    14.9       1
5      7.9    13.4       0
6      7.9    13.5       1

 

The variable ‘success’ is a binary variable that takes the value 1 for individuals who succeeded in gaining admission, and the value 0 for those who did not. Let’s look at the mean values of numeracy and anxiety.

mean(numeracy)
[1] 10.722
mean(anxiety)
[1] 13.954

We begin by fitting a model that includes interactions through the asterisk formula operator. The most commonly used link for binary outcome variables is the logit link, though other links can be used.

model1 <- glm(success ~ numeracy * anxiety, binomial)

glm() is the function that tells R to run a generalized linear model.

Inside the parentheses we give R important information about the model. To the left of the ~ is the dependent variable: success. It must be coded 0 & 1 for glm to read it as binary.

After the ~, we list the two predictor variables. The * indicates that not only do we want each main effect, but we also want an interaction term between numeracy and anxiety.

And finally, after the comma, we specify that the distribution is binomial. The default link function in glm for a binomial outcome variable is the logit. More on that below.

We can access the model output using summary().

summary(model1)
Call:
glm(formula = success ~ numeracy * anxiety, family = binomial)
Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.85712  -0.33055   0.02531   0.34931   2.01048  
Coefficients:
                 Estimate Std. Error z value Pr(>|z|)
(Intercept)       0.87883   46.45256   0.019    0.985
numeracy          1.94556    4.78250   0.407    0.684
anxiety          -0.44580    3.25151  -0.137    0.891
numeracy:anxiety -0.09581    0.33322  -0.288    0.774
(Dispersion parameter for binomial family taken to be 1)
    Null deviance: 68.029  on 49  degrees of freedom
Residual deviance: 28.201  on 46  degrees of freedom
AIC: 36.201

Number of Fisher Scoring iterations: 7

The estimates (coefficients of the predictors – numeracy and anxiety) are now in logits. The coefficient of numeracy is: 1.94556, so that a one unit change in numeracy produces approximately a 1.95 unit change in the log odds (i.e. a 1.95 unit change in the logit).

From the signs of the two predictors, we see that numeracy influences admission positively, but anxiety influences admission negatively.

We can’t tell much more than that as most of us can’t think in terms of logits. Instead we can convert these logits to odds ratios.

We do this by exponentiating each coefficient. (This means raise the value e –approximately 2.72–to the power of the coefficient. e^b).

So, the odds ratio for numeracy is:

OR = exp(1.94556) = 6.997549

However, in this version of the model the estimates are non-significant, and we have a non-significant interaction. Model1 produces the following relationship between the logit (log odds) and the two predictors:

logit(p) = 0.88 + 1.95* numeracy - 0.45 * anxiety - .10* interaction term

The output produced by glm() includes several additional quantities that require discussion.

We see a z value for each estimate. The z value is the Wald statistic that tests the hypothesis that the estimate is zero. The null hypothesis is that the estimate has a normal distribution with mean zero and standard deviation of 1. The quoted p-value, P(>|z|), gives the tail area in a two-tailed test.

For our example, we have a Null Deviance of about 68.03 on 49 degrees of freedom. This value indicates poor fit (a significant difference between fitted values and observed values). Including the independent variables (numeracy and anxiety) decreased the deviance by nearly 40 points on 3 degrees of freedom. The Residual Deviance is 28.2 on 46 degrees of freedom (i.e. a loss of three degrees of freedom).

About the Author: David Lillis has taught R to many researchers and statisticians. His company, Sigma Statistics and Research Limited, provides both on-line instruction and face-to-face workshops on R, and coding services in R. David holds a doctorate in applied statistics.

See our full R Tutorial Series and other blog posts regarding R programming.

 


Generalized Linear Models in R, Part 3: Plotting Predicted Probabilities

July 2nd, 2014 by

In our last article, we learned about model fit in Generalized Linear Models on binary data using the glm() command. We continue with the same glm on the mtcars data set (regressing the vs variable on the weight and engine displacement).

Now we want to plot our model, along with the observed data.

Although we ran a model with multiple predictors, it can help interpretation to plot the predicted probability that vs=1 against each predictor separately.  So first we fit a glm for only (more…)


Generalized Linear Models in R, Part 2: Understanding Model Fit in Logistic Regression Output

June 24th, 2014 by

In the last article, we saw how to create a simple Generalized Linear Model on binary data using the glm() command. We continue with the same glm on the mtcars data set (more…)


Generalized Linear Models in R, Part 1: Calculating Predicted Probability in Binary Logistic Regression

June 18th, 2014 by

Ordinary Least Squares regression provides linear models of continuous variables. However, much data of interest to statisticians and researchers are not continuous and so other methods must be used to create useful predictive models.

The glm() command is designed to perform generalized linear models (regressions) on binary outcome data, count data, probability data, proportion data and many other data types.

In this blog post, we explore the use of R’s glm() command on one such data type. Let’s take a look at a simple example where we model binary data.

(more…)