Like the chicken and the egg, there’s a question about which comes first: run a model or test assumptions? Unlike the chickens’, the model’s question has an easy answer.
There are two types of assumptions in a statistical model. Some are distributional assumptions about the residuals. Examples include independence, normality, and constant variance in a linear model.
Others are about the form of the model. They include linearity and (more…)
by Manolo Romero Escobar
General Linear Model (GLM) is a tool used to understand and analyse linear relationships among variables. It is an umbrella term for many techniques that are taught in most statistics courses: ANOVA, multiple regression, etc.
In its simplest form it describes the relationship between two variables, “y” (dependent variable, outcome, and so on) and “x” (independent variable, predictor, etc). These variables could be both categorical (how many?), both continuous (how much?) or one of each.
Moreover, there can be more than one variable on each side of the relationship. One convention is to use capital letters to refer to multiple variables. Thus Y would mean multiple dependent variables and X would mean multiple independent variables. The most known equation that represents a GLM is: (more…)
In the last post, we examined how to use the same sample when running a set of regression models with different predictors.
Adding a predictor with missing data causes cases that had been included in previous models to be dropped from the new model.
Using different samples in different models can lead to very different conclusions when interpreting results.
Let’s look at how to investigate the effect of the missing data on the regression models in Stata.
The coefficient for the variable “frequent religious attendance” was negative 58 in model 3 and then rose to a positive 6 in model 4 when income was included. Results (more…)
In my last article, Hierarchical Regression in Stata: An Easy Method to Compare Model Results, I presented the following table which examined the impact several predictors have on one’ mental health.
At the bottom of the table is the number of observations (N) contained within each sample.
The sample sizes are quite large. Does it really matter that they are different? The answer is absolutely yes.
Fortunately in Stata it is not a difficult process to use the same sample for all four models shown above.
As I have mentioned previously, Stata stores results in temp files. You don’t have to do anything to cause Stata to store these results, but if you’d like to use them, you need to know what they’re called.
To see what is stored after an estimation command, use the following code:
ereturn list
After a summary command:
return list
One of the stored results after an estimation command is the function e(sample). e(sample) returns a one column matrix. If an observation is used in the estimation command it will have a value of 1 in this matrix. If it is not used it will have a value of 0.
Remember that the “stored” results are in temp files. They will disappear the next time you run another estimation command.
So how do I use the same sample for all my models? Follow these steps.
Using the regression example on mental health I determine which model has the fewest observations. In this case it was model four.
I rerun the model:
regress MCS weeks_unemployed i.marital_status kids_in_house religious_attend income
Next I use the generate command to create a new variable whose value is 1 if the observation was in the model and 0 if the observation was not. I will name the new variable “in_model_4”.
gen in_model_4 = e(sample)
Now I will re-run my four regressions and include only the observations that were used in model 4. I will store the models using different names so that I can compare them to the original models.
My commands to run the models are:
regress MCS weeks_unemployed i.marital_status if in_model_4==1
estimates store model_1a
regress MCS weeks_unemployed i.marital_status kids_in_house if in_model_4==1
estimates store model_2a
regress MCS weeks_unemployed i.marital_status kids_in_house religious_attend if in_model_4==1
estimates store model_3a
regress MCS weeks_unemployed i.marital_status kids_in_house religious_attend income if in_model_4==1
estimates store model_4a
Note: I could use the code if in_model_4 instead of if in_model_4==1. Stata interprets dummy variables as 0 = false, 1 = true.
Here are the results comparing the original models (eg. Model_1) versus the models using the same sample (eg. Model_1a):
Comparing the original models 3 and 4 one would have assumed that the predictor variable “Income level” significantly impacted the coefficient of “Frequent religious attendance”. Its coefficient changed from -58.48 in model 3 to 6.33 in model 4.
That would have been the wrong assumption. That change is coefficient was not so much about any effect of the variable itself, but about the way it causes the sample to change via listwise deletion. Using the same sample, the change in the coefficient between the two models is very small, moving from 4 to 6.
Jeff Meyer is a statistical consultant with The Analysis Factor, a stats mentor for Statistically Speaking membership, and a workshop instructor. Read more about Jeff here.
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.
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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.