Hypothesis Tests with Dependent Variable Transformations |
PROC TRANSREG can also provide approximate tests of hypotheses when the dependent variable is transformed, but the output is more complicated. When a dependent variable has more than one degree of freedom, the problem becomes multivariate. Hypothesis tests are performed in the context of a multivariate linear model with the number of dependent variables equal to the number of scoring parameters for the dependent variable transformation. The transformation regression model with a dependent variable transformation differs from the usual multivariate linear model in two important ways. First, the usual assumption of multivariate normality is always violated. This fact is simply ignored. This is one reason why all hypothesis tests in the presence of a dependent variable transformation should be considered approximate at best. Multivariate normality is assumed even though it is known that the assumption is violated.
The second difference concerns the usual multivariate test statistics: Pillai’s trace, Wilks’ lambda, Hotelling-Lawley trace, and Roy’s greatest root. The first three statistics are defined in terms of all the squared canonical correlations. Here, there is only one linear combination (the transformation), and hence only one squared canonical correlation of interest, which is equal to the R square. It might seem that Roy’s greatest root, which uses only the largest squared canonical correlation, is the only statistic of interest. Unfortunately, Roy’s greatest root is very liberal and provides only a lower bound on the p-value. Approximate upper bounds are provided by adjusting the other three statistics for the one linear combination case. Wilks’ lambda, Pillai’s trace, and Hotelling-Lawley trace are a conservative adjustment of the usual statistics.
These statistics are normally defined in terms of the squared canonical correlations, which are the eigenvalues of the matrix , where is the hypothesis sum-of-squares matrix and is the error sum-of-squares matrix. Here the R square is used for the first eigenvalue, and all other eigenvalues are set to 0 since only one linear combination is used. Degrees of freedom are computed assuming that all linear combinations contribute to the lambda and trace statistics, so the F tests for those statistics are conservative. The p-values for the liberal and conservative statistics provide approximate lower and upper bounds on p. In practice, the adjusted Pillai’s trace is very conservative—perhaps too conservative to be useful. Wilks’ lambda is less conservative, and the Hotelling-Lawley trace seems to be the least conservative. The conservative statistics and the liberal Roy’s greatest root provide a bound on the true p-value. Unfortunately, they sometimes report a bound of 0.0001 and 1.0000.
The following example has a dependent variable transformation and produces Figure 93.73:
title 'Transform Dependent and Independent Variables'; proc transreg data=htex ss2 solve short; model spline(y) = spline(x1-x3); run;
The univariate results match Roy’s greatest root results. Clearly, the proper action is to fail to reject the null hypothesis. However, as stated previously, results are not always this clear.
Transform Dependent and Independent Variables |
Number of Observations Read | 20 |
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Number of Observations Used | 20 |
Spline(y) |
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Algorithm converged. |
Univariate ANOVA Table Based on the Usual Degrees of Freedom | |||||
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Source | DF | Sum of Squares | Mean Square | F Value | Liberal p |
Model | 9 | 110.8822 | 12.32025 | 1.09 | >= 0.4452 |
Error | 10 | 113.2616 | 11.32616 | ||
Corrected Total | 19 | 224.1438 | |||
The above statistics are not adjusted for the fact that the dependent variable was transformed and so are generally liberal. |
Root MSE | 3.36544 | R-Square | 0.4947 |
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Dependent Mean | 0.85490 | Adj R-Sq | 0.0399 |
Coeff Var | 393.66234 |
Adjusted Multivariate ANOVA Table Based on the Usual Degrees of Freedom | |||||
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Dependent Variable Scoring Parameters=3 S=3 M=2.5 N=3 | |||||
Statistic | Value | F Value | Num DF | Den DF | p |
Wilks' Lambda | 0.505308 | 0.23 | 27 | 24.006 | <= 0.9998 |
Pillai's Trace | 0.494692 | 0.22 | 27 | 30 | <= 0.9999 |
Hotelling-Lawley Trace | 0.978992 | 0.26 | 27 | 11.589 | <= 0.9980 |
Roy's Greatest Root | 0.978992 | 1.09 | 9 | 10 | >= 0.4452 |
Univariate Regression Table Based on the Usual Degrees of Freedom | ||||||
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Variable | DF | Coefficient | Type II Sum of Squares |
Mean Square | F Value | Liberal p |
Intercept | 1 | 6.9089087 | 117.452 | 117.452 | 10.37 | >= 0.0092 |
Spline(x1) | 3 | -1.0832321 | 32.493 | 10.831 | 0.96 | >= 0.4504 |
Spline(x2) | 3 | -2.1539191 | 45.251 | 15.084 | 1.33 | >= 0.3184 |
Spline(x3) | 3 | 0.4779207 | 10.139 | 3.380 | 0.30 | >= 0.8259 |
Adjusted Multivariate Regression Table Based on the Usual Degrees of Freedom | |||||||
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Variable | Coefficient | Statistic | Value | F Value | Num DF | Den DF | p |
Intercept | 6.9089087 | Wilks' Lambda | 0.49092 | 2.77 | 3 | 8 | 0.1112 |
Pillai's Trace | 0.50908 | 2.77 | 3 | 8 | 0.1112 | ||
Hotelling-Lawley Trace | 1.036993 | 2.77 | 3 | 8 | 0.1112 | ||
Roy's Greatest Root | 1.036993 | 2.77 | 3 | 8 | 0.1112 | ||
Spline(x1) | -1.0832321 | Wilks' Lambda | 0.777072 | 0.24 | 9 | 19.621 | <= 0.9840 |
Pillai's Trace | 0.222928 | 0.27 | 9 | 30 | <= 0.9787 | ||
Hotelling-Lawley Trace | 0.286883 | 0.24 | 9 | 9.8113 | <= 0.9784 | ||
Roy's Greatest Root | 0.286883 | 0.96 | 3 | 10 | >= 0.4504 | ||
Spline(x2) | -2.1539191 | Wilks' Lambda | 0.714529 | 0.32 | 9 | 19.621 | <= 0.9572 |
Pillai's Trace | 0.285471 | 0.35 | 9 | 30 | <= 0.9494 | ||
Hotelling-Lawley Trace | 0.399524 | 0.33 | 9 | 9.8113 | <= 0.9424 | ||
Roy's Greatest Root | 0.399524 | 1.33 | 3 | 10 | >= 0.3184 | ||
Spline(x3) | 0.4779207 | Wilks' Lambda | 0.917838 | 0.08 | 9 | 19.621 | <= 0.9998 |
Pillai's Trace | 0.082162 | 0.09 | 9 | 30 | <= 0.9996 | ||
Hotelling-Lawley Trace | 0.089517 | 0.07 | 9 | 9.8113 | <= 0.9997 | ||
Roy's Greatest Root | 0.089517 | 0.30 | 3 | 10 | >= 0.8259 |