XLSTAT: ANCOVA analysis in Excel

 

Background

The specificity of Analysis of Covariance (ANCOVA) is that it mixes qualitative and quantitative explanatory variables.

 

Dataset for Example

The (attached) data have been obtained in Lewis T. and Taylor L.R. (1967). Introduction to Experimental Ecology, New York: Academic Press, Inc. They concern 237 children, described by their Gender, Age in months, Height in inches (1 inch = 2.54 cm), and Weight in pounds (1 pound = 0.45 kg).

Using ANCOVA, we want to find out how children weight varies with their gender (a qualitative variable that takes value form), their height and their age, and to verify if a linear model makes sense. The ANCOVA method belongs to a larger family of models called GLM (Generalized Linear Models) as do the  ANOVA .

 

Setting Up

• Open XLSTAT.

• Click on the Modeling data menu and select the ANCOVA feature:

 

• Once you've clicked on the button, the ANCOVA dialog box appears.

In the General tab:

• Select the Weight variable in the Dependent variable field.

• Select the Height and the Age in the Quantitative explanatory variable field.

• Select the Gender variable in the Qualitative variable field.

• Leave the option Variable labels  activated, as we selected the column title for the variables

• The other options have been left at their default value.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

Goodness of fit coefficients

The first table displays the goodness of fit coefficients of the model. The R’² (coefficient of determination) indicates the % of the variability of the dependent variable which is explained by the explanatory variables. The closer to 1 the R’² is, the better the fit.

In this particular case, 63 % of the variability of the Weight is explained by the Height, the Age, and the Gender.

The remainder of the variability is due to some effects (other explanatory variables) that have not been or that could not be measured during this experiment.

We can guess that some genetic and nutritive effects are involved, but it might be that simply by transforming the available variables we could obtain some better results.

 

Variance table

It is important to examine and analyze the variance table (see below). The results enable us to determine whether or not the explanatory variables bring significant information to the model by answer the following questions:

1. Is it valid to use the mean to describe the whole population?

2. Is the information brought by the explanatory variables of value?

First the Fisher's F test is used. 

Given the fact that the probability corresponding to the F value is lower than 0.0001, it means that we would be taking a lower than 0.01% risk in assuming that the null hypothesis is incorrect. We can conclude with confidence then that the three variables do bring a significant amount of information.

 

Type I SS and Type III SS tables

Next we want to find out if the three variables provide the same amount of information. To do this, we examine the Type I SS and Type III SS tables (see below)

The Type I SS table is constructed by adding variables in the model one by one, and evaluating the impact of each on the model sum of squares (Model SS).

In Type I SS, the order in which the variables are selected will influence the results.

The lower the F probability corresponding to a given variable, the stronger the impact of the variable on the model as it is before the variable is added to it.  

We can see from the Type I SS table that Gender brings only a little information to the model, once the Height and the Age have been added.

The Type III SS table is computed by removing one variable of the model at a time to evaluate its impact on the quality of the model.

The order in which the variables are selected will not have any effect on the values in the Type III SS. The Type III SS is generally the best method to use to interpret results when an interaction is part of the model. The lower the F probability corresponding to a given variable, the stronger the impact of the variable on the model.

We can see from the Type III SS Table that Gender actually brings the least information to the model.

 

Parameters Table

The Parameters Table below gives details on the model. It is helpful when predictions are needed, or when you need to compare the coefficients of the model for a given population with the ones obtained for another population.

From the Parameters table, we can see that:

• the p-value for the Gender parameter is 0.83

• the corresponding confidence range includes 0. 

This confirms the weak impact of the Gender on the model. 

If we look at the parameter corresponding to Gender-f, it seems that for a given age and height, being a girl means a small increase of the weight.

 

Residuals Charts

Then, the graph below shows the residuals. These residuals, given the assumptions of the linear regression model, should be normally distributed (95% of the residuals should be in the interval [-1.96, 1.96]).

All values outside this interval are potential outliers, or might suggest that the normality assumption is incorrect. Using XLSTAT's DataFlagger to bring out the residuals that are not in the [-1.96, 1.96] interval, we identify 16 suspicious residuals out of 237 (6% instead of 5% that could lead to rejecting the hypothesis of normality). 

Note: The graph shows the predicted values versus the observed values.

Confidence intervals allow to identify potential outliers.

The residuals bar chart allows us to visualize the standardized residuals versus the Weight. It indicates that the residuals grow with the Weight. The histogram of the residuals enables us to quickly visualize the residuals that are out of the range [-2, 2].

Height, Age, and Gender allow us to explain 63% of the variability of the Weight.

Note: A significant amount of information is not explained by the this model so further analyses of the example would still be necessary.