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Comparing the results of replications in software engineering

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Abstract

Context

It has been argued that software engineering replications are useful for verifying the results of previous experiments. However, it has not yet been agreed how to check whether the results hold across replications. Besides, some authors suggest that replications that do not verify the results of previous experiments can be used to identify contextual variables causing the discrepancies.

Objective

Study how to assess the (dis)similarity of the results of SE replications when they are compared to verify the results of previous experiments and understand how to identify whether contextual variables are influencing results.

Method

We run simulations to learn how different ways of comparing replication results behave when verifying the results of previous experiments. We illustrate how to deal with context-induced changes. To do this, we analyze three groups of replications from our own research on test-driven development and testing techniques.

Results

The direct comparison of p-values and effect sizes does not appear to be suitable for verifying the results of previous experiments and examining the variables possibly affecting the results in software engineering. Analytical methods such as meta-analysis should be used to assess the similarity of software engineering replication results and identify discrepancies in results.

Conclusion

The results achieved in baseline experiments should no longer be regarded as a result that needs to be reproduced, but as a small piece of evidence within a larger picture that only emerges after assembling many small pieces to complete the puzzle.

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Notes

  1. Refer to Camerer et al. (2018) for an overview of the different approaches that have been proposed in different branches of science to check whether or not the results hold across the replications.

  2. We use this example for illustrative purposes, although we are unlikely to ever be able establish this fact, unless we conduct a large enough number of experiments so as to sample the whole population (Ioannidis 2005; Cumming 2013; Camerer et al. 2018).

  3. Cohen’s d\(=\frac {57.49-51.42}{\sqrt {(9.73^{2}+8.30^{2})/2}}=0.67\).

  4. Note that the results of experiments with larger sample sizes are more like the findings for the population.

  5. When we refer to “replications that estimate an identical true effect size”, we do it in the same terms as Borenstein et al. (2011), where the authors refer to a situation in which all factors that could influence the effect size are the same in all the studies, and thus, the true effect size is the same, since all differences in the estimated effects are due to sampling error.

  6. Note that the experimental design does not affect the value of the estimated effect size, it influences the ability to detect the true effect size, which is known as power.

  7. Note, however, that the true effect size is just as important, and, in actual fact, given the problems with questionable research practices and publication bias, a small experiment will most likely overestimate the effect size because it is impossible for a small effect size to be significant.

  8. We are aware that SE data may not be normal (Kitchenham et al. 2017; Arcuri and Briand 2011). However, we opted to use normal distributions as they are a convenient way of expressing the true effect size in the population in terms of Cohen’s d (Borenstein et al. 2011). We decided to express the effect size using Cohen’s d because of its common use in SE (Kampenes et al. 2007). We discuss the shortcomings of simulating normally distributed data in the threats to validity section.

  9. Cohen’s d \(=\frac {58-50}{10}=0.8\) (Cumming 2013).

  10. I2 is interpreted as: 25% low, 50% medium and 75% high (Higgins et al. 2003).

  11. Note, however, that there are other alternatives to random-effects meta-analysis. Empirical Bayes, for instance, has the advantage of being more explicit and using a better approximation algorithm.

  12. For a detailed description of the experiments, their designs, and results please refer to Juristo et al. (2012).

  13. A population of novice testers with limited experience in software development, 12 hours of training on testing techniques, testing toy programs.

  14. For a detailed description of the experiments, their designs, and results please refer to Tosun et al. (2017).

  15. The survey and its results were published elsewhere (Dieste et al. 2017).

  16. For simplicity’s sake, we consider the variables measured throughout the survey as continuous as in Dieste et al. (2017).

  17. This estimate was calculated based on the output of the meta-analysis that we undertook with the metafor R package (Viechtbauer 2010).

  18. τ2 can be estimated with different estimation methods, each of which may provide a potentially different estimate (Langan et al. 2018). In this article, we use restricted maximum likelihood (REML), recommended by Langan et al., applicable to continuous outcomes (Langan et al. 2018). A large number of experiments are needed to estimate precise τ2 parameters (i.e., 5 (Feaster et al. 2011), 10 (Snijders 2011), 15 or even more (McNeish and Stapleton 2016)).

  19. Note that this a sub-optimal approach because of the threat of introducing heterogeneity due to unacknowledged variables. It is better to conduct fewer larger studies. Very often, however, the only option is to run several small studies.

  20. Unfortunately, there are no hard-and-fast rules for establishing how many replications are enough (Borenstein et al. 2011). This is because the precision of the results may be affected by the distribution of sample sizes across the replications (Ruvuna 2004), the experimental design of the replications (Morris and DeShon 2002), the variability of the data (Cumming 2013), etc.

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Acknowledgements

This research was developed with the support of project PGC2018-097265-B-I00, funded by: FEDER/Spanish Ministry of Science and Innovation—Research State Agency.

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Correspondence to Sira Vegas.

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Communicated by: Filippo Lanubile

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Appendices

Appendix A: Probability of reproducing the results of previous experiments

Figure 16 shows the probability of reproducing the results of previous experiments by means of p-values in groups of 2 to 12 replications for a true Cohen’s d of 0.5 at different sample sizes.

Fig. 16
figure 16

Probability that all replications reproduce the results of previous experiments by means of p-values in groups of 2 to 12 replications, all estimating a true Cohen’s of 0.5

Figure 17 shows the probability of reproducing the results of previous experiments by means of p-values in groups of 2 to 12 replications for a true Cohen’s d of 0.8 at different sample sizes.

Fig. 17
figure 17

Probability that all replications reproduce the results of previous experiments by means of p-values in groups of 2 to 12 replications, all estimating a true Cohen’s of 0.8

Appendix B: Joint results of MA vs. overall result of the large-scale experiment

Figure 18 shows the joint estimated effect sizes of meta-analysis for 4, 8 and 12 chunks of sample sizes 36, 18 and 12 each, versus the estimated effect size of the large-scale experiment of sample size 144 at true Cohen’s d of 0.2, 0.5 and 0.8.

Fig. 18
figure 18

Joint result of meta-analysis vs. overall result of the large-scale experiment

Appendix C: Effect size variability when meta-analyzing experiments

Figure 19 shows the distribution of joint effect sizes achieved in 5,000 groups of 1 to 12 identical AB between-subjects replications each with a sample size of 4, meta-analyzed at different true effect sizes (i.e., Cohen’s d of 0.2, 0.5 and 0.8, corresponding to a small, medium and large true effect size, from left to right, respectively).

Fig. 19
figure 19

Meta-analysis: groups with different numbers of replications and true effect sizes and a sample size of 4

Table 10 shows the 95%CIs for the joint effect sizes estimated in Fig. 19.

Figure 20 shows the distribution of joint effect sizes achieved in 5,000 groups of 1 to 12 identical AB between-subjects replications each with a sample size of 100, meta-analyzed at different true effect sizes (i.e., Cohen’s d of 0.2, 0.5 and 0.8, corresponding to a small, medium and large true effect size, from left to right, respectively).

Table 10 95% confidence intervals for joint estimated effect sizes
Fig. 20
figure 20

Meta-analysis: groups with different numbers of replications and true effect sizes and a sample size of 100

Table 11 shows the 95%CIs for the joint effect sizes estimated in Fig. 20.

Table 11 95% confidence intervals for joint estimated effect sizes

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Santos, A., Vegas, S., Oivo, M. et al. Comparing the results of replications in software engineering. Empir Software Eng 26, 13 (2021). https://doi.org/10.1007/s10664-020-09907-7

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