I am currently doing a web scraping study about the online prices of laptops. My objectives are
I have already finished scraping the online prices of laptops last week. I am planning to use a two-way and three-way ANOVA to answer the objectives but as I was checking the assumptions of ANOVA, I have noticed that the homogeneity of variance is violated. I have used Levene's Test in checking the homogeneity of variance. Is it still okay to proceed in using two-way and three-way ANOVA?
Hello Marco Llaguno. Relying on statistical tests of the assumptions for ANOVA (and other types of statistical models) is often not a great idea. In your case, for example, if you have equal sample sizes in all of the cells, ANOVA is quite robust to variance heterogeneity. If you report n, mean and SD for each of the cells in your design, people will be better able to judge whether variance heterogeneity is likely to be problematic.
You mentioned both SPSS and SAS as topics, so I assume you have access to both of them. I'll let others speak about SAS, but if you do need to do something to deal with variance heterogeneity, the ROBUST subcommand that was added to the UNIANOVA command might be helpful. From the Command Syntax reference manual:
"ROBUST displays a table of parameter estimates, along with robust or heteroskedasticity-consistent (HC) standard errors; and t statistics, significance values, and confidence intervals that use the robust standard errors."
https://www.ibm.com/docs/en/spss-statistics/27.0.0?topic=unianova-robust-subcommand-command
HTH.
As Bruce said: the ANOVA is quite robust given the sample sizes are similar.
I think the more relevant question here is if the price difference is really a useful measure, or if actually the price ratio would be the more sensible statistic to analyze. Usually, price changes are proportional, they are also even usually given in percent. Hence, on the absolute scale, variation of prices of expensive laptops are expected to be larger than that of cheaper models. This all becomes relevant when you include laptops with rather different price tags, and it might be negligible when the laptops all cost about the same (but even then I would consider it smarter to analyze relative changes, for purely theoretical reasons and my understanding of the pricing politics and human behaviour [50€ more on a high-end laptop that costs 4000€ is usually considered irrelevant, wheras the same 50€ more for a laptop that is sold for just about 300€ seems inacceptable).
Thank you for your replies. I have scraped data from different websites so I do not have equal sample sizes. I have attached the descriptive statistics of the scraped data.
Thank you Marco Llaguno. Yes, your sample sizes are very variable indeed! So relying on robustness to variance heterogeneity is not a good idea.
Jochen Wilhelm, are you suggesting that Marco use ln(price) as the dependent variable to look at price ratios? E.g.,
https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faqhow-do-i-interpret-a-regression-model-when-some-variables-are-log-transformed/
Cheers,
Bruce
Thank you for your suggestions. I have transformed the data using ln(price); however, I am still unsure if variance heterogeneity is likely to be problematic. Attached are the descriptive statistics of the transformed data.
If homogeneity assumption is not feasible, due to unequal sample sizes and unequal variances, you could apply the Games-Howell test, which yields adequate control over type 1 error and maintains high power. You might refer to Sauder and DeMars (2019), fully cited below, for insights into pairwise-comparison procedures and recommendations.
Sauder, D. C., & DeMars, C. E. (2019). An updated recommendation for multiple comparisons. Advances in Methods and Practices in Psychological Science, 2(1), 26–44. https://doi.org/10.1177/2515245918808784
Good luck,
Hi Marco. One rule of thumb I've seen suggests that when the sample sizes are equal, ANOVA is robust to variance heterogeneity provided the ratio of largest variance to smallest variance is no greater than 4 or 5. I suspect that is a somewhat conservative estimate (when all sample sizes are the same), but it gives you a rough idea of what the smallest:largest ratio can be with equal sample sizes. I don't know if I found the largest and smallest values in your table, but I found variance = .3495 in one cell and variance = .0136 in another cell. And the ratio of those two variances is nearly 26. So I do not see any way to justify and defend use of ordinary ANOVA (with no means of addressing variance heterogeneity) with the original Y or with ln(Y) as the DV (regardless of what Levene's test does or does not show). HTH.
@Bruce, @Jochen I don't understand your use of the term robust in this situation. In what appears to be a similar situation I showed by example that it did make a difference and in fact required a log transformation to set.thimgs . right . See the problem bank.case study for that example. In the attached paper. If I misunderstood something please let me know. Thanks very much, David Booth. BTW Bruce did you ever send the p-value for that K-S test that I requested from you. Let me find the link and I will.post it here shortly. Best wishes, David Booth
@Bruce,. Here's my reply to your other post. Perhaps you should review your posts there. Best wishes David Booth
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