## Nonparametric Analysis of Covariance (Quade's Method)

The below is taken from here. It details how to adjust for covariates in an analysis of covariance when distributions in groups are non-Normal.

While SPSS does not currently offer an explicit option for Quade's rank analysis of covariance, it is quite simple to produce such an analysis in SPSS. 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 will reproduce the F test obtained by Quade for the data on page 1188 of the 1967 JASA paper: Quade, D (1967). Rank analysis of covariance. *Journal of the American Statistical Association* **62** 1187-1200.

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.

It should be noted 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 note that unlike typical parametric ANCOVA analyses, Quade assumed that covariates were random rather than fixed.

Puri & Sen (1969) and Lawson (1983) also mention nonparametric analysis of covariance using ranks.

Other References

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McCann L C, Tomlinson A, Pearce E I, Papa V. (2012). Effectiveness of Artificial Tears in the Management of Evaporative Dry Eye. Cornea. Sep 30. *Cornea.* **31(1)** 1-5.

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Puri M L & Sen P K (1969) Analysis of covariance based on general rank scores *Annals of Mathematical Statistics* **40**, 610-618.

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