8.4a - Equal Slopes Model - using Minitab

Minitab 18

Minitab®

Using our Salary example and the data in the table below, we can run through the steps for the ANCOVA. On this page we will go through the steps using Minitab.

Females Males
Salary years Salary years
80 5 78 3
50 3 43 1
30 2 103 5
20 1 48 2
60 4 80 4
  1. Step 1: Are all regression slopes = 0

    A simple linear regression can be run for each treatment group, Males and Females. (Note: To perform regression analysis on each gender group in Minitab, we will have to sub-divide the salary data manually and separately saving the male data into Male Salary Dataset and female data into Female Salary dataset.

    Running these procedures using statistical software we get the following:

    Males

    Open the Male dataset in the Minitab project file Male Salary Dataset.

    From the menu bar, select Stat > Regression > Regression

    In the pop-up window, select salary into Response and years into Predictors as shown below.

    Minitab dialog box

    Click OK, and here is the output that Minitab displays:

    Regression Analysis: Salary versus years

    The regression equation is

    salary = 24.8 + 15.2 years

    Predictor Coef SE Coef T P
    Constant 24.800 7.534 3.29 0.046
    years 15.200 2.272 6.69 0.007
    S = 7.18331 R-Sq = 93.7% R-Sq(adj) = 91.6%
    Analysis of Variance
    Source DF SS MS F P
    Regression 1 2310.4 2310.4 44.78 0.007
    Residual Error 3 154.8 51.6    
    Total 4 2465.2      
    Females

    Open Minitab dataset Female Salary Dataset.

    From the menu bar select Stat > Regression > Regression

    In the pop-up window, select salary into Response and years into Predictors as shown below.

    Minitab dialog box

    Click OK, and here is the output that Minitab displays:

    Regression Analysis: Salary versus years

    The regression equation is

    salary = 3.00 + 15.0 years

    Predictor Coef SE Coef T P
    Constant 3.000 3.317 0.90 0.432
    years 15.000 1.000 15.00 0.001
    S = 3.16228 R-Sq = 98.7% R-Sq(adj) = 98.2%
    Analysis of Variance
    Source DF SS MS F P
    Regression 1 2250.0 2250.0 225.00 0.001
    Residual Error 3 30.0 10.0    
    Total 4 2280.0      

    In both cases, the simple linear regressions are significant, so the slopes are not = 0.

  2. Step 2: Are the slopes equal?

    We can test for this using our statistical software.

    In Minitab we must now use GLM (general linear model) and be sure to include the covariate in the model. We will also include a ‘treatment x covariate’ interaction term and the significance of this term is what answers our question. If the slopes differ significantly among treatment levels, the interaction p-value will be < 0.05.

    First, open the dataset in the Minitab project file Salary Dataset.

    Then, from the menu select Stat > ANOVA > GLM (general linear model)

    In the dialog box, select salary into Responses and gender into Model, and type gender*years as well.

    Minitab GLM dialog box

    Then, in this dialog box, click on the button "Covariates..." under the text boxes. Select years as Covariates.

    Next, click on the Model box, use the shift key to highlight the gender and years, and then 'add' to create the gender*years interaction:

    Minitab GLM dialog box for model

    Click OK, and the OK again and here is the output that Minitab will display:

    Analysis of Variance

    Source DF Adj SS Adj MS F-Value P-Value
    year 1 4560.20 4560.20 148.06 0.000
    gender 1 216.02 216.02 7.01 0.038
    years*gender 1 0.20 0.20 0.01 0.938
    Error 6 184.80 30.80    
    Total 9 5999.60      
     

    So here we see that the slopes are equal and in a plot of the regressions, we see that the lines are parallel.

    plot
  3. Step 3: Fit an Equal Slopes Model

    We can now proceed to fit an Equal Slopes model by removing the interaction term. This can be easily accomplished by starting again with ANOVA>General Linear Model, but now click on the second item:

    Minitab glm dialog box for model

    Analysis of Variance

    Source DF Adj SS Adj MS F-Value P-Value
    year 1 4560.20 4560.20 172.55 0.000
    gender 1 1254.4 1254.40 47.46 0.000
    Error 7 185.0 26.43    
    Total 9 5999.6      

    To generate the mean comparisons > ANOVA > General Linear Model, but now click on Comparisons.

    minitb glm dialog box for comparisons

    Comparison of salary

    Tukey Pairwise Comparisons: Response = salary, Term = gender

    Grouping information Using the Tukey Method and 95% Confidence

    gender N Mean Grouping
    Male 5 70.4 A
    gender 5 48.0 B

    Means that do not share a letter are significantly different