ANOVA 3: Hypothesis test with F-statistic Analysis of Variance 3 -Hypothesis Test with F-Statistic
ANOVA 3: Hypothesis test with F-statistic
- In the last couple of videos we first figured out the TOTAL variation in these 9 data points right here
- and we got 30, that's our Total Sum of Squares. Then we asked ourselves,
- how much of that variation is due to variation WITHIN each of these groups, versus variation BETWEEN the groups themselves?
- So, for the variation within the groups we have our Sum of Squares within.
- And there we got 6.
- And then the balance of this, 30, the balance of this variation,
- came from variation between the groups, and we calculated it,
- We got 24.
- What I want to do in this video, is actually use this type of information,
- essentially these statistics we've calculated, to do some inferential statistics,
- to come to some time of conclusion, or maybe not to come to some type of conclusion.
- What I want to do is to put some context around these groups.
- We've been dealing with them abstractly right now, but you can imagine
- these are the results of some type of experiment.
- Let's say that I gave 3 different types of pills or 3 different types of food to people taking a test.
- And these are the scores on the test.
- So this is food 1, food 2, and then this over here is food 3.
- And I want to figure out if the type of food people take going into the test really affect their scores?
- If you look at these means, it looks like they perform best in group 3, than in group 2 or 1.
- But is that difference purely random? Random chance?
- Or can I be pretty confident that it's due to actual differences
- in the population means, of all of the people who would ever take food 3 vs food 2 vs food 1?
- So, my question here is, are the means and the true population means the same?
- This is a sample mean based on 3 samples. But if I knew the true population means--
- So my question is: Is the mean of the population of people taking Food 1 equal to the mean of Food 2?
- Obviously I'll never be able to give that food to every human being that could
- ever live and then make them all take an exam.
- But there is some true mean there, it's just not really measurable.
- So my question is "this" equal to "this" equal to the mean 3, the true population of mean 3.
- And my question is, are these equal?
- Because if they're not equal, that means that the type of food given does have some type of impact
- on how people perform on a test.
- So let's do a little hypothesis test here. Let's say that my null hypothesis
- is that the means are the same. Food doesn't make a difference.
- "food doesn't make a difference"
- and that my Alternate hypothesis is that it does. "It does."
- and the way of thinking about this quantitatively
- is that if it doesn't make a difference,
- the true population means of the groups will be the same.
- The true population mean of the group that took food 1 will be the same
- as the group that took food 2, which will be the same as the group that took food 3.
- If our alternate hypothesis is correct, then these means will not be all the same.
- How can we test this hypothesis?
- So we're going to assume the null hypothesis, which is
- what we always do when we are hypothesis testing,
- we're going to assume our null hypothesis.
- And then essentially figure out, what are the chances
- of getting a certain statistic this extreme?
- And I haven't even defined what that statistic is.
- So we're going to define--we're going to assume our null hypothesis,
- and then we're going to come up with a statistic called the F statistic.
- So our F statistic
- which has an F distribution--and we won't go real deep into the details of
- the F distribution. But you can already start to think of it
- as the ratio of two Chi-squared distributions that may or may not have different degrees of freedom.
- Our F statistic is going to be the ratio of our Sum of Squares between the samples--
- Sum of Squares between
- divided by, our degrees of freedom between
- and this is sometimes called the mean squares between, MSB,
- that, divided by the Sum of Squares within,
- so that's what I had done up here, the SSW in blue,
- divided by the SSW
- divided by the degrees of freedom of the SSwithin, and that was
- m (n-1). Now let's just think about what this is doing right here.
- If this number, the numerator, is much larger than the denominator,
- then that tells us that the variation in this data is due mostly
- to the differences between the actual means
- and its due less to the variation within the means.
- That's if this numerator is much bigger than this denominator over here.
- So that should make us believe that there is a difference
- in the true population mean.
- So if this number is really big,
- it should tell us that there is a lower probability
- that our null hypothesis is correct.
- If this number is really small and our denominator is larger,
- that means that our variation within each sample,
- makes up more of the total variation than our variation between
- the samples. So that means that our variation
- within each of these samples is a bigger percentage of the total variation
- versus the variation between the samples.
- So that would make us believe that "hey! ya know... any difference
- we see between the means is probably just random."
- And that would make it a little harder to reject the null.
- So let's actually calculate it.
- So in this case, our SSbetween, we calculated over here, was 24.
- and we had 2 degrees of freedom.
- And our SSwithin was 6 and we had how many degrees of freedom?
- Also, 6. 6 degrees of freedom.
- So this is going to be 24/2 which is 12, divided by 1.
- Our F statistic that we've calculated is going to be 12.
- F stands for Fischer who is the biologist and statistician who came up with this.
- So our F statistic is going to be 12.
- We're going to see that this is a pretty high number.
- Now, one thing I forgot to mention, with any hypothesis test,
- we're going to need some type of significance level.
- So let's say the significance level that we care about,
- for our hypothesis test, is 10%.
- 0.10 -- which means
- that if we assume the null hypothesis, there is
- less than a 10% chance of getting the result we got,
- of getting this F statistic,
- then we will reject the null hypothesis.
- So what we want to do is figure out a critical F statistic value,
- that getting that extreme of a value or greater, is 10%
- and if this is bigger than our critical F statistic value,
- then we're going to reject the null hypothesis,
- if it's less, we can't reject the null.
- So I'm not going to go into a lot of the guts of the F statistic,
- but we can already appreciate that each of these Sum of squares
- has a Chi-squared distribution. "This" has a Chi-squared distribution,
- and "this" has a different Chi-squared distribution
- This is a Chi-squared distribution with 2 degrees of freedom,
- this is a Chi-squared distribution with--And we haven't normalized it and all of that--
- but roughly a Chi squared distribution with 6 degrees of freedom.
- So the F distribution is actually the ratio of two Chi-squared distributions
- And I got this--this is a screenshot from a professor's course at UCLA,
- I hope they don't mind, I need to find us an F table for us to look into.
- But this is what an F distribution looks like.
- And obviously it's going to look different
- depending on the df of the numerator and the denominator.
- There's two df to think about,
- the numerator degrees of freedom and the denominator degrees of freedom
- With that said, let's calculate the critical F statistic,
- for alpha is equal to 0.10,
- and you're actually going to see different F tables for each different alpha,
- where our numerator df is 2, and our denominator df is 6.
- So this table that I got, this whole table is for an alpha of 10%
- or 0.10, and our numerator df was 2 and our denominator
- was 6. So our critical F value is 3.46.
- So our critical F value is 3.46--this value right over here is 3.46
- The value that we got based on our data is much larger than this,
- WAY above it. It's going to have a very, very small p value.
- The probability of getting something this extreme,
- just by chance, assuming the null hypothesis,
- is very low. It's way bigger than our critical F statistic with
- a 10% significance level.
- So because of that we can reject the null hypothesis.
- Which leads us to believe, "you know what, there probably
- IS a difference in the population means."
- Which tells us there probably is a difference in performance
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