David Lillis

Generalized Linear Models in R, Part 5: Graphs for Logistic Regression

August 23rd, 2015 by

In my last post I used the glm() command in R to fit a logistic model with binomial errors to investigate the relationships between the numeracy and anxiety scores and their eventual success.

Now we will create a plot for each predictor. This can be very helpful for helping us understand the effect of each predictor on the probability of a 1 response on our dependent variable.

We wish to plot each predictor separately, so first we fit a separate model for each predictor. This isn’t the only way to do it, but one that I find especially helpful for deciding which variables should be entered as predictors.

model_numeracy <- glm(success ~ numeracy, binomial)
 summary(model_numeracy)
Call:
glm(formula = success ~ numeracy, family = binomial)

Deviance Residuals: 
   Min       1Q   Median       3Q     Max 
-1.5814 -0.9060   0.3207   0.6652   1.8266 

Coefficients:
           Estimate Std. Error z value Pr(>|z|)   
(Intercept) -6.1414     1.8873 -3.254 0.001138 ** 
numeracy     0.6243     0.1855   3.366 0.000763 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

   Null deviance: 68.029 on 49 degrees of freedom
Residual deviance: 50.291 on 48 degrees of freedom
AIC: 54.291

Number of Fisher Scoring iterations: 5

We do the same for anxiety.

model_anxiety <- glm(success ~ anxiety, binomial)

summary(model_anxiety)
Call:
glm(formula = success ~ anxiety, family = binomial)

Deviance Residuals: 
   Min       1Q   Median       3Q     Max 
-1.8680 -0.3582   0.1159   0.6309   1.5698 

Coefficients:
           Estimate Std. Error z value Pr(>|z|)   
(Intercept) 19.5819     5.6754   3.450 0.000560 ***
anxiety     -1.3556    0.3973 -3.412 0.000646 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

   Null deviance: 68.029 on 49 degrees of freedom
Residual deviance: 36.374 on 48 degrees of freedom
AIC: 40.374

Number of Fisher Scoring iterations: 6

Now we create our plots. First we set up a sequence of length values which we will use to plot the fitted model. Let’s find the range of each variable.

range(numeracy)
 [1] 6.6 15.7

range(anxiety)
 [1] 10.1 17.7

 

Given the range of both numeracy and anxiety. A sequence from 0 to 15 is about right for plotting numeracy, while a range from 10 to 20 is good for plotting anxiety.

xnumeracy <-seq (0, 15, 0.01)

ynumeracy <- predict(model_numeracy, list(numeracy=xnumeracy),type="response")

 

Now we use the predict() function to set up the fitted values. The syntax type = “response” back-transforms from a linear logit model to the original scale of the observed data (i.e. binary).

 

plot(numeracy, success, pch = 16, xlab = "NUMERACY SCORE", ylab = "ADMISSION")

lines(xnumeracy, ynumeracy, col = "red", lwd = 2)

image001The model has produced a curve that indicates the probability that success = 1 to the numeracy score.  Clearly, the higher the score, the more likely it is that the student will be accepted.

Now we plot for anxiety.

xanxiety <- seq(10, 20, 0.1)

yanxiety <- predict(model_anxiety, list(anxiety=xanxiety),type="response")

plot(anxiety, success, pch = 16, xlab = "ANXIETY SCORE", ylab = "SUCCESS")

lines(xanxiety, yanxiety, col= "blue", lwd = 2)

image002Clearly, those who score high on anxiety are unlikely to be admitted, possibly because their admissions test results are affected by their high level of anxiety.

****

 

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

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.

 


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.

 


Linear Models in R: Improving Our Regression Model

April 23rd, 2015 by

Stage 2Last time we created two variables and used the lm() command to perform a least squares regression on them, and diagnosing our regression using the plot() command.

Just as we did last time, we perform the regression using lm(). This time we store it as an object M. (more…)


Linear Models in R: Diagnosing Our Regression Model

April 21st, 2015 by

by David Lillis, Ph.D.Stage 2

Last time we created two variables and added a best-fit regression line to our plot of the variables. Here are the two variables again. (more…)


Linear Models in R: Plotting Regression Lines

April 10th, 2015 by

Stage 2Today let’s re-create two variables and see how to plot them and include a regression line. We take height to be a variable that describes the heights (in cm) of ten people. (more…)


R Graphics: Plotting in Color with qplot Part 2

January 13th, 2015 by

In the last lesson, we saw how to use qplot to map symbol colour to a categorical variable. Now we see how to control symbol colours and create legend titles.

M <- structure(list(PATIENT = c("Mary","Dave","Simon","Steve","Sue","Frida","Magnus","Beth","Peter","Guy","Irina","Liz"),
GENDER = c("F","M","M","M","F","F","M","F","M","M","F","F"),
TREATMENT = c("A","B","C","A","A","B","A","C","A","C","B","C"),
AGE =c("Y","M","M","E","M","M","E","E","M","E","M","M"),
WEIGHT_1 = c(79.2,58.8,72.0,59.7,79.6,83.1,68.7,67.6,79.1,39.9,64.7,65.6),
WEIGHT_2 = c(76.6,59.3,70.1,57.3,79.8,82.3,66.8,67.4,76.8,41.4,65.3,63.2),
HEIGHT = c(169,161,175,149,179,177,175,170,177,138,170,165),
SMOKE = c("Y","Y","N","N","N","N","N","N","N","N","N","Y"),
EXERCISE = c(TRUE,FALSE,FALSE,FALSE,TRUE,FALSE,FALSE,TRUE,TRUE,FALSE,FALSE,TRUE),
RECOVER = c(1,0,1,1,1,0,1,1,1,1,0,1)),
.Names = c("PATIENT","GENDER","TREATMENT","AGE","WEIGHT_1","WEIGHT_2","HEIGHT","SMOKE","EXERCISE","RECOVER"),
class = "data.frame", row.names = 1:12)

M

    PATIENT GENDER TREATMENT AGE WEIGHT_1 WEIGHT_2 HEIGHT SMOKE EXERCISE RECOVER
1     Mary      F         A   Y     79.2     76.6    169     Y     TRUE       1
2     Dave      M         B   M     58.8     59.3    161     Y    FALSE       0
3    Simon      M         C   M     72.0     70.1    175     N    FALSE       1
4    Steve      M         A   E     59.7     57.3    149     N    FALSE       1
5      Sue      F         A   M     79.6     79.8    179     N     TRUE       1
6    Frida      F         B   M     83.1     82.3    177     N    FALSE       0
7   Magnus      M         A   E     68.7     66.8    175     N    FALSE       1
8     Beth      F         C   E     67.6     67.4    170     N     TRUE       1
9    Peter      M         A   M     79.1     76.8    177     N     TRUE       1
10     Guy      M         C   E     39.9     41.4    138     N    FALSE       1
11   Irina      F         B   M     64.7     65.3    170     N    FALSE       0
12     Liz      F         C   M     65.6     63.2    165     Y     TRUE       1

Now let’s map symbol size to GENDER and symbol colour to EXERCISE, but choosing our own colours. To control your symbol colours, use the layer: scale_colour_manual(values = c()) and select your desired colours. We choose red and blue, and symbol sizes 3 and 7.

qplot(HEIGHT, WEIGHT_1, data = M, geom = c("point"), xlab = "HEIGHT (cm)", ylab = "WEIGHT BEFORE TREATMENT (kg)" , size = factor(GENDER), color = factor(EXERCISE)) + scale_size_manual(values = c(3, 7)) + scale_colour_manual(values = c("red", "blue"))

Here is our graph with red and blue points:

image001

Now let’s see how to control the legend title (the title that sits directly above the legend). For this example, we control the legend title through the name argument within the two functions scale_size_manual() and scale_colour_manual(). Enter this syntax in which we choose appropriate legend titles:

qplot(HEIGHT, WEIGHT_1, data = M, geom = c("point"), xlab = "HEIGHT (cm)", ylab = "WEIGHT BEFORE TREATMENT (kg)" , size = factor(GENDER), color = factor(EXERCISE)) + scale_size_manual(values = c(3, 7), name="Gender") + scale_colour_manual(values = c("red","blue"), name="Exercise")

image002

We now have our preferred symbol colour and size, and legend titles of our choosing.

That wasn’t so hard! In our next blog post we will learn about plotting regression lines in R.

About the Author:
David Lillis Ph. D. 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.