**ANCOVA Omnibus Test**

**Example 1**: Repeat the analyses of Example 1 of ANOVA Approach to ANCOVA (ANCOVA omnibus test) and Example 1 of Tukey’s HSD for ANCOVA using the Real Statistics **ANCOVA** data analysis tool.

**Figure 1 – Input data**

Referring to the input data in Figure 1, press **Ctrl-m** and double click on the **Analysis of Variance** option (as shown Figure 1 of ANOVA Analysis Tool) in the main dialog box (or click on the **Anova** tab in the multipage interface). Select **ANCOVA** and press **OK** and fill in the dialog box that appears as shown in Figure 2.

**Figure 2 – ANCOVA dialog box**

The output consists of the input and covariate data converted to stacked format in range L3:N38, exactly as is shown in Figure 1. This is followed by the ANCOVA report as shown in Figure 3 and the Tukey HSD output shown in Figure 4.

We note that, although the results shown Figure 3 are similar to those displayed in Figure 7 of ANOVA Approach to ANCOVA, there are some differences in the layout. While the output in Figure 7 of ANOVA Approach to ANCOVA reflects the ANOVA approach to ANCOVA, the output in Figure 3 reflects the regression approach to ANOVA (as described in Regression Approach to ANCOVA).

**Figure 3 – ANCOVA data analysis output**

Whereas the output in Figure 7 of ANOVA Approach to ANCOVA made use only of standard Excel functions, the output in Figure 3 requires use of the following Real Statistics functions.

**Real Statistics Function**: The Real Statistics Resource Pack provides the following array functions where R1 is a range, similar to that in L3:N38 of Figure 1, in stacked format.

**SSAncova**(R1): outputs a column array with* SS _{Covariate}, SS_{Bet} , SS_{W} *and

*SS*

_{Tot}.**AncovaMeans**(R1): outputs a *k* × 3 array with the means and adjusted means for each of the *k* treatment groups.

**AncovaParallel**(R1, *lab*): outputs a 6 × 1 array (or 6 × 2 array including a column of labels if *lab* = TRUE) with the values: R-square of the model which includes the interaction between the tag variables and the covariate variables, R-square of the model excluding these interactions, as well as *df*_{1} , *df*_{2}, *F* and the p-value of the equal slopes test.

Note that Figure 3 contains the array formula =SSAncova(L3:N38) in range Q7:Q9, =AncovaMeans(L3:N38) in range P13:R16 and =AncovaParallel(L3:N38,TRUE) in range P20:Q25.

Note too that the results from the Homogeneity of Slopes Test agrees with the results shown in Figure 4 of ANCOVA Assumptions.

**Tukey’s HSD and Tukey-Kramer**

The Tukey-Kramer analysis output from the ANCOVA data analysis tool is shown in Figure 4 where the contrasts for Method 2 and Method 4 have been filled in. As you can see, the results agree with those shown in Figure 1 of Tukey’s HSD for ANCOVA.

**Figure 4 – Tukey HSD/Kramer data analysis**

Key formulas from Figure 4 are shown in Figure 5 (with reference to cells in Figure 1).

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**Figure 5 – Representative formulas from Figure 4**

Note that by changing the contrast coefficients, it is easy to see that the only pairwise comparisons that yield a significant result are Method 2 vs. Method 3 and Method 2 vs. Method 4.

**Contrasts**

If we had requested **Contrasts** analysis, we would have checked this option in the dialog box shown in Figure 2. Alternatively, we can insert the range L3:N38 as the **Input Range** (leaving the **Covariate Range **empty), uncheck **Column headings included with data** and choose the **Standard **(**stacked**)** format** as the **Input format** as well as the **Contrasts** option. The output is shown in Figure 6 (after filling in the contrast coefficients in range AG7:AG10).

**Figure 6 – Contrast data analysis**

The results are similar to those in Figure 1 of Contrasts for ANCOVA. The formulas in Figure 6 are similar to those in Figure 5. The formula in cell AF15 is

=SQRT(INDEX(SSAncova(L3:N38),3)/(AI11-COUNTA(AF7:AF10)-1) *(SUMPRODUCT(AG7:AG10^2,1/AI7:AI10)+AJ11^2/AK11))