Which of the following best explains how the f ratio examines differences among groups?

Hypotheses: In ANOVA we wish to determine whether the classification (independent) variable affects what we observe on the response (dependent) variable. In the example, we wish to determine whether Temperature affects Learning.

In statistical terms, we want to decide between two hypotheses: the null hypothesis (Ho), which says there is no effect, and the alternative hypothesis (H1) which says that there is an effect.

In symbols:

The t test statistic for two-groups:

The F test statistic for ANOVA:

But variance is difference: It is the average of the differences of a set of values from their mean.

The F-ratio uses variance because ANOVA can have many samples of data, not just two as in T-Tests. Using the variance lets us look at the differences that exist between all of the many samples.

• The numerator
: The numerator (top) of the F-ratio uses the variance between the sample means. If the sample means are all clustered close to each other (small differences), then their variance will be small. If they are spread out over a wider range (bigger differences) their variance will be larger. So the variance of the sample means measures the differences between the sample means.
• The denominator
: The denominator (bottom) of the F-ratio uses the error variance, which is the estimate of the variance expected by chance. The error variance is just the square of the standard error. Thus, rather than using the standard deviation of the error, we use the variance of the error. We do this so that the denominator is in the same units as the numerator.

The most obvious thing about the data is that they are not all the same: The scores are different; they are variable.

The heart of ANOVA is analyzing the total variability into these two components, the mean square between and mean square within. Once we have analyzed the total variability into its two basic components we simply compare them. The comparison is made by computing the F-ratio. For independent-measures ANOVA the F-ratio has the following structure:

Degrees of Freedom: Note that the exact shape depends on the degrees of freedom of the two variances. We have two separate degrees of freedom, one for the numerator (sum of squares between) and the other for the denominator (sum of squares within). They depend on the number of groups and the total number of observations. The exact number of degrees of freedom follows these two formulas (k is the number of groups, N is the total number of observations):

A Conceptual View of ANOVA

We can demonstrate how this works visually. Here are three possible sets of data. In each set of data there are 3 groups sampled from 3 populations. We happen to know that each set of data comes from populations whose means are 15, 30 and 45.

We have colored the data to show the groups. We use

1. Red for the group with mean=15
2. Green for the group with mean=30
3. Blue for the group with mean=45

1. For the first example the populations each have a variance of 4.
   
   
   
   F=854.24, p<.0001.

2. For the second example the outer two populations still have a variance of 4, but the middle one has a variance of 64, so it overlaps the outer two (though they are still fairly well separated).
   
   
   
   F=11.66, p<.0001.

3. For the third example the three populations have a variance of 64, so they all overlap a lot.
   
   
   
   F=1.42, p=.2440.

Post Hoc Tests

When the null hypothesis is rejected you conclude that the means are not all the same. But we are left with the question of which means are different:

Post Hoc tests help give us an answer to the question of which means are different.

1. you reject Ho, and
2. there are 3 or more treatments (groups).

T-Tests can't be used: We can't do this in the obvious way (using T-Tests on the various pairs of groups) because we would get too "rosy" a picture of the significance (for reasons I don't go into). The Post Hoc tests gaurantee we don't get too "rosy" a picture (actually, they provide a picture that is too "glum"!).

Two Post Hoc tests are commonly used (although ViSta doesn't offer any Post Hoc tests):

• Tukey's HSD Test (thats HSD for Honestly Significant Difference). This test can be used only when the groups are all the same size. It determines a single value that is the minimum difference between a pair of groups that is needed for the difference to be significant at a specific alpha level.
• Scheffe's Test is very conservative. It involves computing an F-Ratio that has a numerator that is a mean-square that is based on only the two groups being compared (the denominator is the regular error variance term).

Example

1. State the Hypotheses
:
The hypotheses, for ANOVA, are:




2. Set the Decision Criterion

We arbitrarily set


3. Gather the Data
:
The data are obtained from 60 subjects, 20 in each of 3 different experimental conditions. The conditions are a Placebo condition, and two different drug conditions. The independent (classification) variable is the experimental condition (Placebo, DrugA, DrugB). The dependent variable is the time the stimulus is endured.

Here are the data as shown in ViSta's data report:



The data may be gotten from the ViSta Data Applet. Then, you can do the analysis that is shown below yourself.

4. Visualize the Data

We visualize the data and the model in order to see if the assumptions underlying the independent-measures F-test are met. The assumptions are:
1. The observations within each sample must be independent (this assumption is satisfied by the nature of the experimental design).
2. The populations from which the samples are selected must be normal (the data and model visualizations can inform us about this).
3. The populations from which the samples are selected must have equal variance (the data and model visualizations can inform us about this also). This is called homogeneity of variance.

The data visualization is shown below. The boxplot shows that there is somewhat more variance in the "DrugA" group, and that there is an outlier in the "DrugB" group. The Q plots (only the "DrugB" Q-Plot is shown here) and the Q-Q plot show that the data are normal, except for the outlier in the "DrugB" group.



5. Evaluate the Null Hypothesis

We use ViSta to calculate the observed F-ratio, and the observed probability level. The report produced by ViSta is shown below. The information we want is near the bottom:



We note that F=4.37 and p=.01721. Since the observed p < .05, we reject the null hypothesis and conclude that it is not the case that all group means are the same. That is, at least one group mean is different than the others.

Here is the F distribution for df=2,57 (3 groups, 60 observations). I have added the observed F=4.37:



6. Visualize the Model

Finally, we also visualize the ANOVA model to see if the assumptions underlying the independent-measures F-test are met. The boxplots are the same as those for the data. The partial regression plot shows that the model is significant at the .05 level of significance, since the curved lines cross the horizontal line. The residual plot shows the outline in the "DrugB" group, and shows that the "DrugA" group is not as well fit by the ANOVA model as the other groups. Here is the model visualization:

What does the F test measure quizlet?

Which of the following in a t

Which of the following can be assessed using a repeated measures t

What type of hypothesis states that one particular groups average is higher than the other groups?