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Problem background: As part of my research, I have written two algorithms that can select a set of features from a data set (gene expression data from cancer patients). These features are then tested to see how well they can classify an unseen sample as either cancer or non-cancer. For each run of the algorithm, a solution (a set of features) is generated and tested on Z unseen samples. Percentage accuracy of the solution is calculated like this: (correct classifications / Z) * 100.

I have two algorithms: algorithm X & algorithm Y

I have three separate (different cancers) data sets: data set A, data set B & data set C. These data sets are very different from each other. They don't have the same number of samples or same number of measurements (features) per sample.

I have run each algorithm 10 times on each data set. So, algorithm X has 10 results from data set A, 10 from data set B and 10 from data set C. Overall, algorithm X has 30 results.

My problem: I would like to see if algorithm X's combined performance across all three data sets is statistically significantly different from algorithm Y's combined performance.

Is it possible for me to combine results for algorithm X from each data set into a single set of results? This way, I would have 30 standardized results for algorithm X and 30 standardized results for algorithm Y. I can then use the t-test to see if there is a significant difference between the two methods.

Edit - These are Evolutionary Algorithms, so they return a slightly different solution each time they are run. However, if there's a feature in a sample that when present can strongly classify the sample as either being cancer or non-cancer, then that feature will be selected almost every time the algorithm is run.

I get slightly different results for each of the 10 runs due to the following reasons:

  • These algorithms are randomly seeded.
  • I use repeated random sub-sampling validation (10 repeats).
  • The datasets that I use (DNA microarray and Proteomics) are very difficult to work with in the sense that there are many local optima the algorithm can get stuck in.
  • There are lots of inter-feature and inter-subset interactions that I would like to detect.
  • I train 50 chromosomes and they are not restricted to any particular length.  They are free to grow and shrink (although selection pressure guides them towards shorter lengths).  This also introduces some variation to the final result.

Having said, the algorithm almost always selects a particular subset of features!

Here's a sample of my results (only 4 runs out of 10 for each algorithm is shown here):

Dataset/run     Algorithm X     Algorithm Y
A 1             90.91           90.91
A 2             90.91           95.45
A 3             90.91           90.91
A 4             90.91           90.91

B 1             100             100
B 2             100             100
B 3             95.65           100
B 4             95.65           86.96

C 1             90.32           87.10
C 2             70.97           80.65
C 3             96.77           83.87
C 4             80.65           83.87

As you can see, I've put together results for two algorithms from three datasets. I can do Kruskal-Wallis test on this data but will it be valid? I ask this because:

  • I'm not sure accuracies in different data sets are commensurable. If they are not, then putting them together like I've done would be meaningless and any statistical test done on them would also be meaningless.
  • When you put accuracies together like this, the overall result is susceptible to outliers. One algorithm's excellent performance on one dataset may mask it's average performance on another dataset.

I cannot use t-test in this case either, this is because:

  • Commensurability - the t-test only makes sense when the differences over the data sets are commensurate.
  • t-test requires that the differences between the two algorithms compared are distributed normally, there's no way (at least that I'm aware of) to guarantee this condition in my case.
  • t-test is affected by outliers which skew the test statistics and decrease the test’s power by increasing the estimated standard error.

What do you think? How can I do a comparison between algorithm X & Y in this case?

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  • $\begingroup$ Do your algorithms involve some kind of randomness, or why else do you run them 10 times each on every dataset? $\endgroup$ Commented Feb 6, 2013 at 9:15
  • $\begingroup$ Yes! They are Evolutionary Algorithms (stochastic algorithms). So, each time they produce a different result. However, if there are strong features (genes/a single value from a sample), then they are selected more often than not. The aim of the algorithm is to select those genes that are strongly correlated to a particular class (e.g. Ovarian cancer) so that they can be used in early diagnosis/prediction of cancer in the future. $\endgroup$ Commented Feb 6, 2013 at 10:49

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Unless your algorithms have huge differences in performance and you have huge numbers of test cases, you won't be able to detect differences by just looking at the performance.

However, you can make use of apaired design:

  • run all three algorithms on exactly the same train/test split of a data set, and
  • do not aggregate the test results into % correct, but keep them at the single test case level as correct or wrong.

For the comparison, have a look at McNemar's test. The idea behind exploiting a paired design here is that all cases that both algorithms got right and those that both got wrong do not help you distinguishing the algorithms. But if one algorithm is better than the other, there should be many cases the better algorithm got right but not the worse, and few that were predicted correctly by the worse method but wrong by the better one.

Also, Cawley and Talbot: On Over-fitting in Model Selection and Subsequent Selection Bias in Performance Evaluation, JMLR, 2010, 1, 2079-2107. is highly relevant.

Because of the random aspects of your algorithms, you'll also want to check the same split of the same data set multiple times. From that you can estimate the variation between different runs that are otherwise equal. It may be difficult to judge how different the selected sets of variables are. But if your ultimate goal is predictive performance,then you can also use the variation between predictions of the same test case during different runs to measure the stability of the resulting models.

You'll then also want to check (as indicated above) variation due to different splits of the data set and put this into relation with the first variance.

Fractions (like % correctly recognized samples) are usually assumed to be distributed binomially, in certain cases a normal approximation is possible, but the small-print for this hardly ever met in fields with wide data matrices. This has the consequence that confidence intervals are huge for small numbers of test cases. In R, binom::binom.confint calculates confidence intervals for the true proportion given no. of tests and no. of successes.

Finally, My experience with genetic optimization for spectroscopic data (my Diplom thesis in German) suggests that you should check also the training errors. GAs tend to overfit very fast, arriving at very low training errors. Low training errors are not only overoptimistic, but they also have the consequence that the GA cannot differentiate between lots of models that seem to be equally perfect. (You may have less of a problem with that if the GA internally also randomly subsamples train and internal test sets).

Papers in English:

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  • $\begingroup$ Thank you for an excellent analysis! I will have a go at a paired design. You are spot on regarding over fitting. Next phase of my research is going to concentrate on avoiding over fitting while training. This is really important as my end objective is to produce an algorithm that is fully automated so that oncologists can use for early diagnosis of cancers. I'm really interested in reading your paper but I'm afraid I cannot read German. Please let me know if there is an English version. Thank you again for your detailed input. $\endgroup$ Commented Feb 7, 2013 at 18:18
  • $\begingroup$ @revolusions: The information is a bit spread out over some papers. But I added a list with 2 about the GA variable selection, and one about uncertainty in measuring classification error rates. If you have questions (or don't have access to the papers), send me an email. $\endgroup$ Commented Feb 7, 2013 at 19:43
  • $\begingroup$ Thank you! I managed to download the last paper but can't seem to get the first two. I will try from my university computer tomorrow and will let you know if I manage to download them. $\endgroup$ Commented Feb 7, 2013 at 20:06
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You are running featuer selection with GA 10 times and every time you get a different output !!

First, If you start by the same seed, you should always get the same selected featuer subset. However, If you are using a random seed, also, most probably, you should get almost the same selected features. One reason for getting the same selected featuer is stated in your post. Also, for a fair comparison, you may use the same seeds in the runs of A for the B's experiments.

Second, you may use cross-validation or bootstraping for comparison. This way you get a more representative comparison. In this case, there is a source of variation i.e. random training samples which seems stronger than random seeds. Thus, the comparison may reveal which algorthim is really better.

Finally, you may use t-test as you proposed or directly use some non-parametric tests like Kruskal-Wallis test.

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  • $\begingroup$ Thank you so much for your answer! I'd like to clarify a few things and maybe get your opinion afterwards again as I'm still confused about the comparison, hope you can help! You said: <blockquote>Also, for a fair comparison, you may use the same seeds in the runs of A for the B's experiments</blockquote> This is a very good point, thank you. I've edited my question to address the points you raised. It'll be great if you can have a look and let me know what you think. $\endgroup$ Commented Feb 7, 2013 at 13:14
  • $\begingroup$ You may do a separate comparison between classifiers for every dataset. Initially, check mean and standard deviation. If for all datasets, one classifier beats the other. Then, we are done. Otherwise, you may use an "ensemble" idea. Check the new sample, if it belongs to dataset A use the better classifier and so and so forth. However, there might be some statistical test based on paired view that I am not aware about for this case. $\endgroup$
    – soufanom
    Commented Feb 7, 2013 at 18:09
  • $\begingroup$ Thank you again for your input. I have done what you suggested and there is no clear winner. This is why I decided to put everything together and see if there would be a winner. This "ensemble" idea sounds interesting. Is there any place where I can read more about this? Thanks for your help. $\endgroup$ Commented Feb 7, 2013 at 18:26
  • $\begingroup$ One more thing, consider comparing sensitivity and specifity. For the ensemble source, check the book attached at my post stats.stackexchange.com/questions/47075/… $\endgroup$
    – soufanom
    Commented Feb 8, 2013 at 5:02

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