TwoSample3

Summary

Highlights

• Homogeneity of variance is vital for t-test assumptions and results. 🎯
• Unequal variances can mislead interpretations of mean differences. 🤔
• Bootstrap and randomization help manage assumption violations. 🔧
• Empirical t-statistics distributions help determine rejection regions. 🚦

Key Takeaways

• The homogeneity of variance assumption is crucial for accurate t-test results. 📊
• If group variances are not similar, the t-test results could be misleading. 🚨
• Bootstrapping and randomization methods can act as remedies for violations of assumptions. 🔄
• Understanding empirical t-distributions can help in setting accurate thresholds. 📈

Overview

Chapters

• 00:00 - 01:00: Introduction to Homogeneity and Variances The chapter begins with the introduction of homogeneity and variance, highlighting its significance as an assumption in hypothesis testing, specifically for the t-test. While the t-test is generally robust, it is sensitive to violations of the homogeneity of variance assumption. The chapter likely delves into theoretical hypothesis tests and the necessity of these underlying assumptions.
• 01:00 - 02:00: Assumptions of the t-test The chapter discusses the assumptions critical to the t-test, focusing primarily on the assumption of homogeneity of variance. It explains that the t-test, while robust against violations of many of its assumptions, is sensitive to violations of this specific assumption. Homogeneity of variance means that the variances of the two groups being compared (Group 1 and Group 2) must be similar and not statistically different. To satisfy this assumption, the populations from which the samples are drawn should have equal variances, which is necessary for the validity of the t-test results.
• 02:00 - 03:00: The Components of a t-test Statistic The chapter explains the concept of pooling variance estimates in the context of calculating a t-test statistic. It emphasizes that the denominator of the t statistic includes normalized variance estimates from two populations or samples. For these estimates to be combined, they must be similar enough so that any differences are only attributable to sampling error.
• 03:00 - 04:00: Interpreting Significant Differences The chapter 'Interpreting Significant Differences' explains the details of the t-test, which is a statistical test used to determine if there are significant differences between the means of two groups. It discusses the components of the t-test, which include the numerator and the denominator. The numerator is the population mean difference under the null hypothesis, expected to be zero unless there is an actual effect, which would be indicated by a difference in means between Group 1 and Group 2. The denominator is the estimate of noise or sampling error, also known as the pooled variance. The chapter also touches upon interpreting the test statistic, which relates to understanding the results of the t-test.
• 04:00 - 05:00: Example: Variances and Mean Differences This chapter deals with understanding the implications of variances and mean differences, especially in the context of T distributions. It highlights the issue that arises when the assumption of homogeneity of variance is violated, which creates uncertainty in determining the factor responsible for differences in outcomes. The chapter mentions that if the variance estimates of two groups are notably similar, any observed differences are likely due to the mean differences between the groups.
• 05:00 - 06:00: Understanding t-statistics and Sampling Error The chapter focuses on understanding t-statistics and sampling error. It discusses the process of rejecting the null hypothesis when a statistically significant difference between sample means is found. However, it warns that if there is a large variance between the two samples, the significant difference might not be interpretable. This is because the observed differences might be due to variance rather than actual differences between population means.
• 06:00 - 07:00: Issues with Unequal Variances The chapter titled 'Issues with Unequal Variances' discusses the problems that arise when there is an extreme value of T due to sampling error, which may lead to misinterpretation of P values. It explains that under the null hypothesis, we assume that the two sample means are the same, originating from the same population, and generated through the same process. This raises concerns when analyzing any score in either sample in the presence of unequal variances.
• 07:00 - 08:00: Solutions for Unequal Variances The chapter discusses the concept of addressing unequal variances in statistical samples. It highlights that if the null hypothesis is true, individual scores could belong to either sample without affecting the outcome, as both samples would be representative of the same population. The example of a score of 27 is used to illustrate that under the null hypothesis, the allocation of this score to either of two groups is arbitrary, as they originate from the same population.
• 08:00 - 09:00: Empirical Distribution of t-statistics The chapter titled 'Empirical Distribution of t-statistics' discusses the challenges in averaging variances from different populations when there are significant differences in their variance. This discrepancy can impede the ability to contextualize mean differences based on sampling error. The explanation highlights the fundamental concept of the t-statistic, which is the difference between means divided by the expected sampling error. To illustrate this concept, the chapter suggests using a simple example with a hypothetical sample.
• 09:00 - 10:00: Conclusion: Handling Violations in t-tests The chapter discusses handling violations in t-tests, focusing on the comparison of two samples with different variances. It describes the visual representation of two samples, where sample one (in red) has a lower variance compared to sample two (in blue), which has a much greater variance. This difference in variance between the two samples is a key point of discussion in handling t-test assumptions and potential violations.

TwoSample3 Transcription

• 00:00 - 00:30 so it's worth addressing the idea of homogeneity variants a little bit more specific like it's a really important assumption for the t-test and the t-test is generally speaking pretty robust but it's really sensitive to violations of this assumption so homogeneity of variants I'll start by telling you that most theoretical hypothesis tests require certain assumptions
• 00:30 - 01:00 and although the t-test is pretty robust against violations of its assumptions it's pretty sensitive to violations of this assumption so homogeneity variance remember means that the variance of Group 1 and the variance of group 2 are pretty similar to one another or not statistically different to meet this assumption two samples for two populations from which the samples arise must have equal variances and this is needed to justify
• 01:00 - 01:30 pooling their variance estimates so remember we said that the denominator of the T statistic Inc included a normalized estimate of the variance in population one or sample one plus a normalized estimate of the variance in the second or for the second group or sample and so in order to add those two together they need to be similar enough that only the only differences between them are due to sampling error
• 01:30 - 02:00 so the t-test has two elements it has its numerator which is the value for the population mean difference under the null hypothesis that would be zero but if you have an effect it might be that the mean of group one is bigger than the mean of group two or vice versa and then we also have the denominator element which is the our noise estimate um and that's the value for the pooled variance so the sampling error we know that if the test statistic falls
• 02:00 - 02:30 into our rejection region so it's in the more extreme ends of the of our T distribution the problem if we violate the homogeneity of variance assumption is that we don't know which of these values is responsible for the outcome if the pooled variance if the two variance estimates the variance estimate of Group 1 and group two are very similar to one another then we know that any differences have to be due to the difference between
• 02:30 - 03:00 their means and that allows us to reject the null hypothesis if we find a statistically significant difference however if we find a statistically significant difference and the variance in the two samples is highly different to one another then it's we then we can't really interpret that significant difference because we don't know whether there's actually really a difference between the population means or if it's really any differences we're seeing are really being caused by this variance so
• 03:00 - 03:30 an extreme value of T might simply reflect sampling error due to chance and that becomes more likely and then we can't interpret our P values accurately can we so let's look a little close more closely at this so under the null hypothesis we assume that the two sample means are the same but they came from the same population that the process by which the two samples were generated is the same and this means that any score in either
• 03:30 - 04:00 sample could just as easily have been found in the other sample right if the null hypothesis is true and I get a score of 27. doesn't matter whether that 27 could easily be in group 1 or group two it doesn't matter I could flip a coin and put that 27 in one of the groups heads it's group one Tails its group two it doesn't matter if the null hypothesis is true because if the null hypothesis is true those two samples came from the exact same population
• 04:00 - 04:30 makes sense now large differences in the variance of those populations defeat our ability to average those variances and therefore to contextualize the mean differences according to how much sampling error we've got so let's look at an example so we know that t is the difference between the means divided by the expected sampling error so let's pretend and this is a baby like toy example here so let's pretend that we have sample one
• 04:30 - 05:00 printed in red and we have a second sample here sample two in blue and what you can see from these two samples you can see that they each have a mean and the means are reasonably far apart and they each have a distribution but you can also see that the distribution the width of this the variance of this distribution is much greater in Sample two than it is in Sample one so let's say in Sample one we have this much variance and in Sample two we have
• 05:00 - 05:30 this much variance and we'll put them over here next to one another for reference so we can see the mean of sample one and the mean of sample two here and when we add them together we're going to get some kind of a sampling error estimate that is there that is a sum of their of these two variances now let's look at our next difference so we've added them together we have ours our average our estimate of the variance now so that's called our pooled variance or
• 05:30 - 06:00 our standard error of the differences now let's look at our mean difference our mean difference is this big here if we put it over here for reference what do you notice think about that very carefully does this mean difference look similar to the pooled variants or different from the pole variants yeah it looks pretty similar so now we have this problem because these are unequal so it becomes very difficult to tell
• 06:00 - 06:30 whether this difference is due to chance now in this if we got this ratio here a t-test that had this mean difference and this pulled variance our T would be pretty close to uh we wouldn't be able to tell we would not get a statistically significant result there even though the mean difference between the groups here is pretty big if both the samples had this amount of variance right if if sample two had the same variance assemble one
• 06:30 - 07:00 we would certainly see that our mean difference is significantly bigger than our sampling error because our pooled variance would be smaller so because we are creating where T is a ratio between the difference or of the difference between the means divided by that expected sampling error we need to be careful that the sampling error is roughly identical across the two samples otherwise when we make our comparison we
• 07:00 - 07:30 get a non-significant result so what do we do if that happens well one of the ways to do that is to instead of using our theoretical T distribution to look up our critical value we can build a distribution of T statistics empirically we can simulate it or we can use a kind of randomization method for like bootstrap resampling or there are a couple of other randomization methods that we tend to use so remember that if the null hypothesis is
• 07:30 - 08:00 true any score but can be in either sample so if we Shuffle the scores randomly from group one to group two or vice versa we can define a function that randomly assigns each of the scores to one of the groups it's not resampling it's more like the shuffling we did when we did the gender difference version and then we have random assignment to groups without without replacement there so we randomly assign all of our scores to new groups we calculate the T statistic and we save that calculated T statistic to an array and we will repeat
• 08:00 - 08:30 this process of course lots and lots of times and that distribution that we build up which is a t distribution so it's a distribution of T statistics now so now we're building up a new distribution it doesn't contain means it doesn't contain standard deviations it contains actual T statistics because we've calculated T and that's the score we're saving to our array this empirical distribution of T is used to determine our 95 thresholds or like we're a rejection region boundaries lie
• 08:30 - 09:00 so that's what we do if you have a violation of your homogeneity variance or any of the other assumptions with t-tests and that's all for this lecture