### Troubleshooting

## Problem

I want to run a rank analysis of covariance, as discussed in: Quade, D. (1967). Rank analysis of covariance. Journal of the American Statistical Association, 62(320), 1187-1200. Can SPSS produce this analysis?

## Resolving The Problem

Recent versions of SPSS Statistics include a Python Essentials-based extension to perform Quade's nonparametric ANCOVA and pairwise comparisons among groups. In the menus see Analyze>Nonparametric Tests>Quade Nonparametric ANCOVA.

Without access to the extension, it is still fairly simple to perform the basic analysis in the program. The required steps are as follows:

1) Rank the dependent variable and any covariates, using the default settings in the SPSS RANK procedure. This is done for all cases, ignoring the grouping variable.

2) Run a linear regression of the ranks of the dependent variable on the ranks of the covariates, saving the (raw or Unstandardized) residuals, again ignoring the grouping factor.

3) Run a one-way analysis of variance (ANOVA), using the residuals from the regression in the prior step as the dependent variable, and the grouping variable as the factor. The F test resulting from this ANOVA is the F statistic Quade used.

Note that Quade actually proposed centering the ranks for each of the ranked variables by subtracting their means, and performing the linear regression without an intercept. However, the residuals produced by ignoring these two steps are the same, so the method discussed here is a simpler way to get to the same final results.

The following commands reproduce the F test obtained by Quade for the data on page 1188 of the 1967 JASA paper:

1) Rank the dependent variable and any covariates, using the default settings in the SPSS RANK procedure. This is done for all cases, ignoring the grouping variable.

2) Run a linear regression of the ranks of the dependent variable on the ranks of the covariates, saving the (raw or Unstandardized) residuals, again ignoring the grouping factor.

3) Run a one-way analysis of variance (ANOVA), using the residuals from the regression in the prior step as the dependent variable, and the grouping variable as the factor. The F test resulting from this ANOVA is the F statistic Quade used.

Note that Quade actually proposed centering the ranks for each of the ranked variables by subtracting their means, and performing the linear regression without an intercept. However, the residuals produced by ignoring these two steps are the same, so the method discussed here is a simpler way to get to the same final results.

The following commands reproduce the F test obtained by Quade for the data on page 1188 of the 1967 JASA paper:

```
data list list / group y x1 x2.
begin data
1 16 26 12
1 60 10 21
1 82 42 24
1 126 49 29
1 137 55 34
2 44 21 17
2 67 28 2
2 87 5 40
2 100 12 38
2 142 58 36
3 17 1 8
3 28 19 1
3 105 41 9
3 149 48 28
3 160 35 16
end data.
rank variables=y x1 x2.
regression dep=Ry
/enter Rx1 Rx2
/save resid.
oneway RES_1 by group.
```

Note that the assumptions made by Quade (see page 1187) include that the distribution of any covariates is the same in each group, so the utility of the method is restricted to situations where groups are equivalent on any covariates. Also, unlike typical parametric ANCOVA analyses, Quade assumed that covariates were random rather than fixed.

## Related Information

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## Historical Number

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### Document Information

**Modified date:**

14 June 2022

## UID

swg21477497