One question that seems to come up pretty often is:
What is the difference between logistic and probit regression?
Well, let’s start with how they’re the same:
Both are types of generalized linear models. This means they have this form:
One question that seems to come up pretty often is:
Well, let’s start with how they’re the same:
Both are types of generalized linear models. This means they have this form:
Proportion and percentage data are tricky to analyze.
Much like count data, they look like they should work in a linear model.
They’re numerical. They’re often continuous.
And sometimes they do work. Some proportion data do look normally distributed so estimates and p-values are reasonable.
But more often they don’t. So estimates and p-values are a mess. Luckily, there are other options. (more…)
I received a question recently about R Commander, a free R package.
R Commander overlays a menu-based interface to R, so just like SPSS or JMP, you can run analyses using menus. Nice, huh?
The question was whether R Commander does everything R does, or just a small subset.
Unfortunately, R Commander can’t do everything R does. Not even close.
But it does a lot. More than just the basics.
So I thought I would show you some of the things R Commander can do entirely through menus–no programming required, just so you can see just how unbelievably useful it is.
Since R commander is a free R package, it can be installed easily through R! Just type install.packages("Rcmdr") in the command line the first time you use it, then type library("Rcmdr") each time you want to launch the menus.
Import data sets from other software:
Define Numerical Variables as categorical and label the values
Open the data sets that come with R packages
Merge Data Sets
Edit and show the data in a data spreadsheet
Personally, I think that if this was all R Commander did, it would be incredibly useful. These are the types of things I just cannot remember all the commands for, since I just don’t use R often enough.
Yes, R Commander does many of the simple statistical tests you’d expect:
What is surprising though, is how many higher-level statistics and models it runs:
In other words–you can use R Commander to run in R most of the analyses that most researchers need.
A sample of the types of graphs R Commander creates in R without you having to write any code:
The nice part is that it does not only do simple versions of these plots. You can, for example, add regression lines to a scatter plot or run histograms by a grouping factor.
If you’re ready to get started practicing, click here to learn about making scatterplots in R commander, or click here to learn how to use R commander to sample from a uniform distribution.
In my last blog post we fitted a generalized linear model to count data using a Poisson error structure.
We found, however, that there was over-dispersion in the data – the variance was larger than the mean in our dependent variable.
In my last couple of articles (Part 4, Part 5), I demonstrated a logistic regression model with binomial errors on binary data in R’s glm() function.
But one of wonderful things about glm() is that it is so flexible. It can run so much more than logistic regression models.
The flexibility, of course, also means that you have to tell it exactly which model you want to run, and how.
In fact, we can use generalized linear models to model count data as well.
In such data the errors may well be distributed non-normally and the variance usually increases with the mean values.
As with binary data, we use the glm() command, but this time we specify a Poisson error distribution and the logarithm as the link function.
The natural log is the default link function for the Poisson error distribution. It works well for count data as it forces all of the predicted values to be positive.
In the following example we fit a generalized linear model to count data using a Poisson error structure. The data set consists of counts of high school students diagnosed with an infectious disease within a period of days from an initial outbreak.
cases <-
structure(list(Days = c(1L, 2L, 3L, 3L, 4L, 4L, 4L, 6L, 7L, 8L,
8L, 8L, 8L, 12L, 14L, 15L, 17L, 17L, 17L, 18L, 19L, 19L, 20L,
23L, 23L, 23L, 24L, 24L, 25L, 26L, 27L, 28L, 29L, 34L, 36L, 36L,
42L, 42L, 43L, 43L, 44L, 44L, 44L, 44L, 45L, 46L, 48L, 48L, 49L,
49L, 53L, 53L, 53L, 54L, 55L, 56L, 56L, 58L, 60L, 63L, 65L, 67L,
67L, 68L, 71L, 71L, 72L, 72L, 72L, 73L, 74L, 74L, 74L, 75L, 75L,
80L, 81L, 81L, 81L, 81L, 88L, 88L, 90L, 93L, 93L, 94L, 95L, 95L,
95L, 96L, 96L, 97L, 98L, 100L, 101L, 102L, 103L, 104L, 105L,
106L, 107L, 108L, 109L, 110L, 111L, 112L, 113L, 114L, 115L),
Students = c(6L, 8L, 12L, 9L, 3L, 3L, 11L, 5L, 7L, 3L, 8L,
4L, 6L, 8L, 3L, 6L, 3L, 2L, 2L, 6L, 3L, 7L, 7L, 2L, 2L, 8L,
3L, 6L, 5L, 7L, 6L, 4L, 4L, 3L, 3L, 5L, 3L, 3L, 3L, 5L, 3L,
5L, 6L, 3L, 3L, 3L, 3L, 2L, 3L, 1L, 3L, 3L, 5L, 4L, 4L, 3L,
5L, 4L, 3L, 5L, 3L, 4L, 2L, 3L, 3L, 1L, 3L, 2L, 5L, 4L, 3L,
0L, 3L, 3L, 4L, 0L, 3L, 3L, 4L, 0L, 2L, 2L, 1L, 1L, 2L, 0L,
2L, 1L, 1L, 0L, 0L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 0L, 0L,
0L, 1L, 1L, 0L, 0L, 0L, 0L, 0L)), .Names = c("Days", "Students"
), class = "data.frame", row.names = c(NA, -109L))
attach(cases)
head(cases)
Days Students
1 1 6
2 2 8
3 3 12
4 3 9
5 4 3
6 4 3
The mean and variance are different (actually, the variance is greater). Now we plot the data.
plot(Days, Students, xlab = "DAYS", ylab = "STUDENTS", pch = 16)
Now we fit the glm, specifying the Poisson distribution by including it as the second argument.
model1 <- glm(Students ~ Days, poisson) summary(model1) Call: glm(formula = Students ~ Days, family = poisson) Deviance Residuals: Min 1Q Median 3Q Max -2.00482 -0.85719 -0.09331 0.63969 1.73696 Coefficients: Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.990235 0.083935 23.71 <2e-16 ***
Days -0.017463 0.001727 -10.11 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 215.36 on 108 degrees of freedom
Residual deviance: 101.17 on 107 degrees of freedom
AIC: 393.11
Number of Fisher Scoring iterations: 5
The negative coefficient for Days indicates that as days increase, the mean number of students with the disease is smaller.
This coefficient is highly significant (p < 2e-16).
We also see that the residual deviance is greater than the degrees of freedom, so that we have over-dispersion. This means that there is extra variance not accounted for by the model or by the error structure.
This is a very important model assumption, so in my next article we will re-fit the model using quasi poisson errors.
****
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.
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)
The 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)
Clearly, 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.