I have to warn you here that I made this list up. I've never heard of "translation" validity before, but I needed a good name to summarize what both face and content validity are getting at, and that one seemed sensible. All of the other labels are commonly known, but the way I've organized them is different than I've seen elsewhere. Let's see if we can make some sense out of this list. First, as mentioned above, I would like to use the term construct validity to be the overarching category. Construct validity is the approximate truth of the conclusion that your operationalization accurately reflects its construct.

All of the other terms address this general issue in different ways. Second, I make a distinction between two broad types: translation validity and criterion-related validity. In translation validity , you focus on whether the operationalization is a good reflection of the construct. This approach is definitional in nature -- it assumes you have a good detailed definition of the construct and that you can check the operationalization against it. In criterion-related validity , you examine whether the operationalization behaves the way it should given your theory of the construct.

This is a more relational approach to construct validity. If all this seems a bit dense, hang in there until you've gone through the discussion below -- then come back and re-read this paragraph. Let's go through the specific validity types. I just made this one up today!

### Why Predictive Models Performance Evaluation is Important

See how easy it is to be a methodologist? I needed a term that described what both face and content validity are getting at. In essence, both of those validity types are attempting to assess the degree to which you accurately translated your construct into the operationalization, and hence the choice of name. Let's look at the two types of translation validity.

In face validity , you look at the operationalization and see whether "on its face" it seems like a good translation of the construct. This is probably the weakest way to try to demonstrate construct validity. For instance, you might look at a measure of math ability, read through the questions, and decide that yep, it seems like this is a good measure of math ability i.

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- Comparing and predicting between several methods of measurement!

Or, you might observe a teenage pregnancy prevention program and conclude that, "Yep, this is indeed a teenage pregnancy prevention program. Note that just because it is weak evidence doesn't mean that it is wrong. We need to rely on our subjective judgment throughout the research process. It's just that this form of judgment won't be very convincing to others. We can improve the quality of face validity assessment considerably by making it more systematic.

For instance, if you are trying to assess the face validity of a math ability measure, it would be more convincing if you sent the test to a carefully selected sample of experts on math ability testing and they all reported back with the judgment that your measure appears to be a good measure of math ability. In content validity , you essentially check the operationalization against the relevant content domain for the construct. This approach assumes that you have a good detailed description of the content domain, something that's not always true.

For instance, we might lay out all of the criteria that should be met in a program that claims to be a "teenage pregnancy prevention program. Then, armed with these criteria, we could use them as a type of checklist when examining our program. A positive difference of residuals refers to a better prediction of procedure 1 with respect to procedure 2.

Suppose that split 1 results in a positive difference for sample j. When we are interested in the joint null distribution over the 2 splits, we need to be able to compute the conditional probability of a positive difference in the second split given the positive difference in the first one. In addition, residuals of test samples overlapping between splits are also dependent. We believe that 2.

Inequality 2. Finally, dropping all assumptions on P , we obtain the following bound. In particular, it equals 0. Our testing procedure is summarized by the following algorithm. We assume that the median p -value is used as the summary p -value. Under H 0 , each S j is equally likely positive and negative. Therefore, compute the p -value p i by sign-permutations. Repeat 2. We performed a simulation study 1 to verify our conjecture that the most liberal bounds, 2.

We used a setting with 2 sparse linear models. The number of available covariates is of the same order as the number of samples.

When applying the one-split approach, one would like to know the consistency of the decision with respect to repeated splitting. In the simulation, we have estimated this consistency as well. We observed that the multisplit approach using c 0. Second, we observed that the one-split approach can be more powerful on average than the multisplit approaches but only in the very low power region when its results are inconsistent. This is mainly due to splits for which the alternative is more pronounced.

We consider 2 applications of our inference procedure. We focus on survival as a response variable. All testing is one-sided. The first data set is the microarray data set of Bullinger and others which can be downloaded from the GEO data base accession number GSE It consists of gene expression profiles from patients with acute myeloid leukemia and contains expression values of genes.

Thirteen samples with more than missing values are removed. The resulting data set contains expression profiles of genes. Remaining missing values are imputed using K -nearest neighbors. Overall survival is used as the end point in the analysis. This is a high-dimensional setting for which the number of covariates, p , is larger than N. These comparisons all evaluate several methods by several prediction error metrics.

Here, we first focus on the comparison between lasso regression and ridge regression. We used implementations of lasso and ridge as described in Park and Hastie and Van Houwelingen and others , respectively. The nice variable selection property of lasso motivates our asymmetric preference for the 2 prediction procedures. If ridge regression predicts significantly better than lasso regression for our data, it may be worthwhile to consider using this more expensive procedure. Figure 1 a displays the histogram of p -values as determined from the 50 splits.

Figure 1 b shows the potential effect of the split fraction on the prediction error. This implies that one should be careful with deciding between 2 procedures based on cross-validated prediction errors for one given split fraction. In conclusion, there is no statistical evidence to prefer ridge over lasso in this application, so the lasso is recommended because of its additional feature selection property.

Lasso versus ridge: a histogram of p -values and b prediction errors for several split fractions; median solid and quartiles dashed. A second application of our test is illustrated on a second data set of 99 Dutch patients with acute myeloid leukemia. At diagnosis, DNA expression was measured for 33 genes and its relation with survival was examined Hess and others In addition, the degree of methylation was assessed for 25 different regions. Here, we compare a lasso regression that contains both predictor sets versus one containing only the gene expression markers. If the procedure with both sets predicts significantly better than the procedure with DNA expression alone, it may be considered for prognosis of patients.

Otherwise, the less costly alternative without methylation assessments is preferred. We increased I to because we were not satisfied with the precision of the estimates of the summary p -values from 50 splits. Figure 2 a displays the histogram of p -values. Hence, there is substantial evidence that the methylation markers are crucial in addition to the mRNA markers for predicting survival in this data set.

Methylation versus no methylation: a histogram of p -values and b prediction errors for several split fractions; median solid and quartiles dashed. Our testing procedure is based on simple CV. We have also considered 3 utilizations of the bootstrap for the purpose of inference, but none was found to be appropriate.

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Below, we discuss these bootstrapping schemes and why these fail for our aim. First, in this splitting setting, bootstrapping the residuals in a multivariate fashion is not feasible due to the complex dependencies between residuals of test samples which are in each other's training set for 2 different splits. Hence, exchangeability of the samples is not guaranteed.

## The Energetic Cost of Walking: A Comparison of Predictive Methods

Second, it has been shown before that estimation of the prediction error may be improved by combining bootstrap CV with the apparent error rate Schumacher and others Bootstrap CV resamples a training set of size N from the original sample and uses samples not present in the bootstrap as test samples. This results in a somewhat pessimistic estimate since the bootstrap sample does not contain all the information contained in the original sample. Therefore, a weighted average with the apparent error error within the training sample is computed. Unfortunately, this elegant approach seems inappropriate in an inference procedure.

First, the residuals contributing to the apparent error cannot be used for testing our null hypothesis. Second, the bootstrapping procedure results in test samples of different size, hence containing different amounts of independent information. Combining these into one inference is not straightforward.

Finally, one could try to generate a confidence interval, for example, for the median residual, by resampling the entire data set several times. This confidence interval would then be used as an alternative to our testing procedure. Such a procedure would be computationally intensive.

More importantly, we are interested in comparing the predictive performance of the 2 procedures trained on a large proportion q of the data.

Hence, the confidence interval would not reflect the uncertainty under our null of interest, which is concerned about predictors using independent data in the training phase. We have illustrated the use of our method for comparing 2 prediction procedures in several settings. In this setting, an alternative testing approach would be to use multiple random permutation of the response variables and to reapply the prognostic method to each permutation. If the covariates contain any predictive value, then the observed prediction error should be in the left tail of the resulting permutation distribution of prediction error.

However, in high-dimensional settings this approach is computationally cumbersome since one needs many permutations to obtain a reasonable estimate of the p -value and each permutation requires a new training loop. Permutation could also be used in a setting with 2 sets of covariates: fix the response and the corresponding covariate values of set 1, while permuting the labels of the covariates of set 2. However, this approach may be invalid. The null hypothesis implies that the second covariate set is not associated with the response but does not imply it to be nonassociated with the covariates in the first set, which is a necessary condition implying every permutation to be equally likely under H 0 for the resulting p -value to be valid.

The optimal choice of the ratio of the training and test set sizes in the proposed procedure depends on the data set under consideration. Ideally one wants to approach N as much as possible by n , so that the inference on a prediction based on n samples can be extrapolated to N. This is a well-known phenomenon: the prediction error is bounded by the amount of information the covariates contain with respect to the response Dobbin and Simon However, this is a computationally intensive task since it requires evaluation of both procedures under multiple split fractions.

While the multivariate normal assumption in Theorems 1 and 2 is difficult to verify, it may be useful to have an asymptotic justification, when the number of samples is large. The asymptotic results may heavily depend on the type of residuals as well as the prediction procedures considered. Hence, such a study requires a focus on particular prediction paradigms and procedures.

## Comparing and predicting between several methods of measurement.

Another extension of our method is to the comparison of more than 2 predictors. This would imply using a nonparametric K -sample test that accounts for blocking, such as the Friedman test, instead of the Wilcoxon signed-rank test. The theory on combining p -values from several splits may then be applied without any change. A unique characteristic of our method is the testing framework targeting on prediction. We connect 2 well-accepted concepts in applied statistics: prediction error rates and p -values as a measure of evidence against the null hypothesis.

The procedure is entirely nonparametric, hence it is robust against outliers and may also be used for nonlikelihood prediction methods. In fact, it may be used in any prediction or classification setting in which it is possible to define a residual of the predicted value with respect to the observation. The method owes much of its power to the pairing principle in the trainingâ€”test splitting. Moreover, combining p -values avoids arbitrariness related to the study of a single split. In conclusion, we suggest that our method may play an important role in the evaluation of competitive prediction procedures and sets of prognostic factors in many studies.

We thank Jelle Goeman for providing the implementation of ridge regression to us. Conflict of Interest: None declared. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. Testing the prediction error difference between 2 predictors Mark A. Oxford Academic. Google Scholar.