ANOVA Flashcards
In words and symbols, what is the null hypothesis in one-way ANOVA? In words and symbols, what is the alternative hypothesis in one-way ANOVA?
H0: µ1 = µ2 = . . . = µk (The population means of the k groups are all equal.) Ha: µi does not equal µj for at least one i, j pair. (The population means of the k groups are not all equal.)
is sample means are very different then SST is
large, if sample means similar SST is small *looking at variability between groups
What is SSE
variability WITHIN groups,
what is mean square
sum of squares/dof *Sp^2=MSE
what is the F statistic
ratio of mean square treatment/mean square error (last two columns)
what happens of H0 is false
MST tends to be bigger than MSE and the test statistic tends to be large *greater value of F statistic the greater evidence against null
how do u find Sp^2
MSE
determine number of pairwise comparisons
(# of groups/2) ** doing that n!/x!(n-x)! thing
In a one-way ANOVA the F statistic is found to be 0.00002. Does this value give strong evidence against the null hypothesis?
No. In one-way ANOVA, only large values of the F test statistic give strong evidence against the null hypothesis.
- Under what conditions would the test statistic be equal to 0? - Under what conditions would the test statistic be equal to 1?
- The test statistic will equal 0 if MST = 0 (provided MSE 6= 0). - MST will equal 0 when there is no variability in the sample means—when the k groups all have the same value of the sample mean
. Consider the boxplots in Figure 1, representing 3 separate and independent samples of size 20. Which of the following statements are true?
(a) If we performed a one-way ANOVA on these 3 samples, then the p-value would be small.
(b) If we carried out a t-test of H0: µA = µC against a two-sided alternative, then the p-value would be small.
(c) The use of the ANOVA procedures for the test of H0: µA = µB = µC would be a bad idea, as the assumptions are clearly violated.
(d) If we carried out a pooled-variance t-test of H0: µA = µC against a two-sided alternative, and a one-way ANOVA F test of H0: µA = µC , then the test statistics would have the relationship: t 2 = F.
(e) If we carried out a pooled-variance t-test of H0: µA = µC against a two-sided alternative, then a one-way ANOVA of the test of H0: µA = µC , the p-values would be exactly equal.
(a) True. Visually there is very, very strong evidence against H0, implying a small p-value .
(b) True. Visually there is very, very strong evidence against H0, implying a small p-value .
(c) False. There is no visual evidence against the assumptions of normality and a common population variance.
(d) True. For two-sample problems the pooled-variance t procedure and one-way ANOVA are equivalent tests, with t 2 = F.
(e) True. For two-sample problems the pooled-variance t procedure and one-way ANOVA are equivalent tests, resulting in the exact same p-value.
one-way ANOVA, with 10 observations in each of 5 different groups. Suppose the null hypothesis is true, and the assumptions are true.
(a) What is the distribution of the test statistic?
(b) What is the distribution of the p-value?
(c) What would the p-value equal on average?
(a) The test statistic will have an F distribution with 4 df in the numerator, and 45 df in the denominator. If the null hypothesis and the assumptions are true, the F test statistic has an F distribution with k − 1 degrees of freedom in the numerator and n − k degrees of freedom in the denominator. If there are 10 observations in each of 5 groups, there are 5 − 1 = 4 degrees of freedom in the numerator, and 50 − 5 = 45 degrees of freedom in the denominator.
(b) If the null hypothesis and the assumptions are true, then the p-value will have a uniform distribution between 0 and 1.
(c) Since the p-value is uniformly distributed between 0 and 1, the p-value will equal 0.5 on average (if the null hypothesis and the assumptions are true).
(a) If the null hypothesis (and the assumptions) are true, then the test statistic in one-way ANOVA has an F distribution.
true
(b) In one-way ANOVA, we assume that the observations within each group are normally distributed, and that all groups have the same population variance.
true
(c) In one-way ANOVA, we assume that the observations within each group are normally distributed, and that all groups have the same population mean.
false
(d) The test statistic in one-way ANOVA can be negative.
false, F test stat is a ratio of variances which can never be negative
(e) If the null hypothesis is false, then MST will tend to be bigger than MSE.
true
If the null hypothesis is true, then the F statistic will be infinite
false
If the null hypothesis is true, then the F statistic will equal 1
False. If H0 is true, the F statistic will have a median that is near 1, but it can take on any value between 0 and ∞.
If the null hypothesis is false, then the F statistic will sometimes be less than one
True. The F statistic can take on any nonnegative value, regardless of whether the null hypothesis is true or false
If the null hypothesis is false, then the F statistic will sometimes be less than 0.
False. The F statistic cannot be negative.
If the null hypothesis is false, then the p-value will be less than 0.05
False. If the null hypothesis is false, we can still end up with a large p-value.
If MSE = 0, then the p-value will equal the F statistic.
false
If the population means are equal, then the F statistic will equal 1.
False. If the population means are equal, then H0 is true and the F statistic will have an F distribution
If the sample means are all equal, then the F statistic will equal 1.
False. If the sample means are equal, the F statistic will equal 0
If the sample means are all equal, then the null hypothesis is true
False. If the sample means are all equal, then there is absolutely no evidence whatsoever against H0. But that still does not mean it must be true.
The mean square error is the pooled variance.
True.
If the population means are very different, we expect the F statistic to be greater than one.
True. If H0 is true (the population means are equal), then the F statistic will have a median that is somewhere around 1. If H0 is false, the F statistic will tend to be larger than 1.
We will reject the null hypothesis at the 5% level whenever the sample means are all equal
False. When the sample means are all equal there is absolutely no evidence whatsoever against the null hypothesis
An assumption of one-way ANOVA is that the groups all have different population variances.
False. One-way ANOVA assumes the population variances are equal.
If the assumptions of the pooled-variance t procedures are met, then the t test and the F test are equivalent tests, with t 2 = F.
true
The equal variance assumption is not important for large sample sizes, due to the central limit theorem.
. False. A violation of the equal variance assumption can be very problematic, even for large sample sizes. The central limit theorem can help out with a violation of the normality assumption.
what is MSE and MST mean
variabily within groups, MST is variability of group means
