The Analysis Factor
Statistical Consulting, Resources, and Statistics Workshops for Researchers
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:
[Read more…] about The Difference Between Logistic and Probit Regression
An incredibly useful tool in evaluating and comparing predictive models is the ROC curve.
Its name is indeed strange. ROC stands for Receiver Operating Characteristic. Its origin is from sonar back in the 1940s; ROCs were used to measure how well a sonar signal (e.g., from an enemy submarine) could be detected from noise (a school of fish).
In its current usage, ROC curves are a nice way to see how any predictive model can distinguish between the true positives and negatives. [Read more…] about What Is an ROC Curve?
When you have data measuring the time to an event, you can examine the relationship between various predictor variables and the time to the event using a Cox proportional hazards model.
In this webinar, you will see what a hazard function is and describe the interpretations of increasing, decreasing, and constant hazard. Then you will examine the log rank test, a simple test closely tied to the Kaplan-Meier curve, and the Cox proportional hazards model.
Note: This training is an exclusive benefit to members of the Statistically Speaking Membership Program and part of the Stat’s Amore Trainings Series. Each Stat’s Amore Training is approximately 90 minutes long.
I received a question recently about R Commander, a free R package. R Commander is the powerhouse of our upcoming workshop R for SPSS Users.
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.
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.
by David Lillis, Ph.D.
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.
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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.
by David Lillis, Ph.D.
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 survival 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 – 1.0* 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).
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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.