General linear models are a family of statistical models based on the normal distribution.

A lot of classical statistical methods come under the unifying umbrella of general linear models:

• analysis of variance (ANOVA),
• linear regression,
• t-tests,
• analysis of covariance (ANCOVA)

are four examples of statistical methods which are all examples of general linear models.

General linear models are often called linear models for short.

General linear models have parameters, like all statistical models.

Fitting a general linear model means using the data to find suitable values for the parameters of the model.

You can use the lm() function in R to fit a general linear model. Below is an example:

		  # Fit a simple general linear model to the human height data
		  m = lm(HEIGHT ~ 1, data=human)
	      

The lm() function uses a maximum likelihood approach to fit the model, but we don't need to know about the details of the fitting. We will need to know how to interpret the results of the fitting process.

In texts you will see several names used for essentially the same concept. Here are a few that you will encounter in this lesson:

• Error equivalent to residual
• Fitted value equivalent to the prediction from a model
• Explanatory variable equivalent to independent variable or predictor variable
• Response variable equivalent to dependent variable or predicted variable

A statistical glossary on Brightspace has a more complete list of statistical terminology

The R code below uses the lm() command to fit and display the first general linear model shown in the video above.

		  # Subset data for heights of women
		  humanF = subset(human, SEX=='F')
		  
		  # Fit a general linear model to heights of women
		  m = lm(HEIGHT ~ 1, data=humanF)

		  # Produce a summary of the fitted model
		  summary(m)		  
	      

The R code below uses the lm() command to fit and display the second general linear model shown in the video above.

		  # Fit a general linear model to heights of women and men
		  m = lm(HEIGHT ~ 1 + SEX, data=human)

		  # Produce a summary of the fitted model
		  summary(m)		  
	      

The R code below uses the lm() command to fit and display the general linear model shown in the video above.

		  # Subset data for heights of men
		  humanM = subset(human, SEX=='M')
		  
		  # Fit a general linear model to the relationship
		  # between HEIGHT and WEIGHT of men
		  m = lm(HEIGHT ~ 1 + WEIGHT, data=humanM)

		  # Produce a summary of the fitted model
		  summary(m)
	      

The R code below uses the lm() command to fit and display the general linear model shown in the video above.

		  # Fit a general linear model to the relationship between
		  # height and weight for women and men
		  m = lm(HEIGHT~1+SEX+WEIGHT+SEX:WEIGHT, data=human)

		  # Produce a summary of the fitted model
		  summary(m)
	      

A general linear model makes a number of assumptions about the population it is trying to model.

The main assumptions are (in order of decreasing importance):

The R code below uses the lm() command to fit and display the general linear model shown in the video above.

		  # Fit a general linear model to the relationship between
		  # height and weight for women and men
		  m = lm(HEIGHT~1+SEX, data=human)

		  # Produce residuals versus fitted plot (homogeneity of variance)
		  plot(m, which=1)

		  # Produce QQ plot of the residuals (normality)
		  plot(m, which=2)
	      

R will produce four validation plots by default

		  # Display the default four validation plots
		  plot(m)
	      

The assumptions of homogeneity of variance and normality can be validated by looking at the residuals from a fitted general linear model.

General linear models are fairly robust to mild violations of the assumptions. They are most robust to departures from normality.

Below we give some examples of residual versus fitted plots and quantile-quantile plots from fitted general linear models that suggest one of these two assumptions has been violated.

This example fits a general linear model with two means (i.e. two fitted values)

This quantile-quantile plot (right) shows that the residuals are right-skewed. The normality assumption is violated.

This example fits a general linear model with two means (i.e. two fitted values)

This quantile-quantile plot (right) shows that the residuals are under-dispersed. The normality assumption is violated.

This example fits a general linear model with two means (i.e. two fitted values)

The residual versus fitted plot (left) shows more variability around one mean compared to the other. The homogeneity of variance assumption is violated. The normality assumption is also questionable.

This example fits a straight line relationship

The residual versus fitted plot (left) shows a funnel-shaped pattern in the residuals (variability increases with increasing fitted values). The homogeneity of variance assumption is violated.

This example fits a straight line relationship

The residual versus fitted plot (left) shows a curved 'U-shaped' pattern in the residuals (red line). The homogeneity of variance assumption is violated because the relationship in the data is not linear.

This example fits a straight line relationship

The residual versus fitted plot (left) shows a curved 'n-shaped' pattern in the residuals (red line). The homogeneity of variance assumption is questionable because the relationship in the data is not linear.