Calculating Fobt for all possible combinations of variances for each sample Calculating the probability of (F) for each different value of Fobt

Transcription

1 PSY 305 Module 5-A AVP Transcript During the past two modules, you have been introduced to inferential statistics. We have spent time on z-tests and the three types of t-tests. We are now ready to move to the next type of statistical analysis; the analysis of variance. In this module, we will examine the one-way ANOVA. With z-tests and t-tests, we have used the mean to test the null hypothesis. ANOVA is an analysis of variance so we will use variance rather than means to test the hypothesis. ANOVA is also called the F-test and is named after Fisher who developed the test. In ANOVA, we will calculate F-obtained and compare this value to F-critical. As with t-tests, we also have a sampling distribution of F. The distribution gives all possible F values along with the probability of F for each value, assuming sampling is random from the population. This distribution is generated empirically by: Taking all possible samples of size n 1 and n 2 Estimating the population variance from each of the samples using s 1 2 and s 2 2 Calculating Fobt for all possible combinations of variances for each sample Calculating the probability of (F) for each different value of Fobt So, when do you use a One-way ANOVA? You will use this test if you have one IV with two or more levels. Thus, if you want to study the effects of caffeine on stress, caffeine is the IV and stress is the DV. Let s say that participants are assigned to 0 cups, 2 cups, or 4 cups of coffee. This IV has three levels so a one-way ANOVA would be appropriate. ANOVA and independent t-tests are similar in that both can be used for one IV with two levels. However, a t-test can be used for only two levels, while an ANOVA can be used with more than two levels. T tests uses means to test the null while ANOVA uses variance. F can never be negative because all values are squared. A t can be negative or positive. The F distribution is essentially a folded over t distribution so t squared = F. Here you can see the t and F distributions. Recall that a t-test can be used for a one or two tailed test. An F test, however, can only be a one-tailed test. Positive and negative differences end up in the right tail because everything is squared. To summarize, both an independent t-test and one-way ANOVA can be used for one IV with two levels. However, if you have three or more levels, you must use the one-way ANOVA. Let s take a moment to test what you ve learned so far. If you conduct an analysis of variance and find a negative F, it means that: A) it is probably significantly different from 1 B) it is probably a Type I error C) it is probably a Type II error D) you have made a mistake in your calculations 1

2 Let s see how you did. The answer is D you have made a mistake in your calculations. Recall that ANOVA is a one-tail test and must always be positive. If you get a negative number, the calculation is incorrect. Now that you have some background information regarding ANOVA, let s begin examining how to calculate an ANOVA. An ANOVA tests the ratio of between-group variability to within-group variability. This table distinguishes between the between group and within group variability. The between group variance gives us the difference between the group means, is the treatment effect, and is the variance that we can explain. We can explain this variance because it is due to the manipulation of our IV. The within group variance is the differences between peoples scores within each group. It is the random error or the variance we cannot explain. The F statistics is calculated by examining F = variability between groups variability within groups Or, in other words: F = variance we can explain variance we can t explain The F-statistic is based on a one-tailed distribution. Therefore, F-tests are always one-tailed. The null states that the groups are equal or nothing happened. H 0 = M 1 = M 2 The alternative hypothesis is always that H0 is not true the groups are different or unequal. H 1 = M 1 M 2 Recall that we used a t-table to find the critical t when using t-tests for hypothesis testing. We will now use the F-table to find the F critical value. This process is similar to t-tests, but now we have to look up two different degrees of freedom to find our critical value. This table is found in the back of your textbook. To find F critical, you need to calculate the degrees of freedom between and degrees of freedom within. The df between is the denominator while the df within is the numerator. In the table, for each degree of freedom, alpha =.01 is listed on the first row and alpha =.05 is listed on the second row. Let s examine how to calculate each degree of freedom. For degrees of freedom, we need to take an -1. A = the total number of groups and n = the number in each group. Degrees of freedom between is the degrees of freedom for the treatment group. To find df between, simply take a -1, or the number of groups minus 1. 2

3 Finally, degrees of freedom within is the degrees of freedom for the error term. To find this value, take n-1 then multiply by a. Recall that df between + df within = df total. Let s take a look at an example. A researcher wants to know what type of therapy is helps people with a Simple Phobia the most. She assigns 15 people to 1 of 3 groups so that there are 5 people in each group: a psychoanalysis group, a cognitive therapy group, and a behavioral therapy group. After therapy, she has each person rate the severity of their phobia from 0 (it s gone) to 10 (it s the worst it has ever been). Which test is most appropriate? If you said a one-way ANOVA, you are absolutely right! Take a look at the data. As you can see, there are three levels of the IV: Psycho- Analysis Cognitive Therapy Behavioral Therapy The last row shows the means for each level of the IV. As previously stated, in this example we have one IV with three levels. Let s consider our hypotheses for this example. Recall that the null states that the groups are equal. So, H0 = M1 = M2 = M3 or the mean for psychoanalysis = the mean for cognitive therapy = the mean for behavioral therapy. The alternative hypothesis states that the groups are different so H1 = M1 M2 M3 or the mean for psychoanalysis does not equal the mean for cognitive therapy does not equal the mean for behavioral therapy. When we calculate an ANOVA, there are several important steps to follow: 1. Find df 2. Calculate Fobs 3. Look up Fcrit 4. Compare Fobs to Fcrit to see if you have a significant effect Remember that: F = variability between groups divided by variability within groups So, to find F, we must calculate the between-group variability and the within-group variability It is also a good idea to always create an ANOVA summary table to help you keep up with the pieces of information you calculate. You will need to calculate sum of squares, df, and the mean square (MS) for the between and within effect in order to calculate F obtained. This table organizes the data and assists with the calculation. In order to calculate F-obtained, we must calculate the variance. In ANOVA, variance or variability is called the Mean Square (or MS for short) So F = MSbetween/ MSwithin 3

4 MS = SS/df There are three SSs: SStotal = SSbetween = (also called SSA or SSeffect) SSwithin = (also called SSS/A or SSerror) You need to calculate sum of squares and degrees of freedom first so that you can get the MS between and MS error terms. Remember that (x x-bar)2 = sum of squares SS/n 1 = variance variance = standard deviation These formulas were covered in Psy 205 So, once we have our SS, we can divide by our degrees of freedom to get our MS between and MS error. Then we divide MSbetween over MSerror to get our F-obtained. Let s examine how to calculate ANOVA using the computational method. First, you need to retrieve the document called Computational method ANOVA from the Doc sharing area. Be sure to follow along on your handout and take notes since you will have an assignment very similar to this one. In the table, you will find the data for two levels of the independent variable (A1 and A2). Under each column of data, you will find the mean for A1 and A2 as well as the sum of X and the sum of X squared. Recall that in order to get the sum of X squared, you must square each number in the column and then add them together. Now let s get our sum of squares total. First, we need to add the sum of x squared. We need to add 154 plus 400 since we have two groups. Then we divide T square by an. A = the number of groups while N = the number in each group. T = the sum of X for A1 plus the sum of X for A2. Now we subtract to get 554 minus 507 = 47. This is the sum of squares total. Next we need to find the sum of squares for our between effect or SSa. Notice that the second portion of the formula is T squared divided by an. We will just insert this number (507) from the above step. The first part of the formula is sum of A squared divided by n. To get A squared, we need to square the sum of X for A 1 and A 2. We get 30 squared plus 48 squared and divide by 6 which equals 534. To get sum of squares between, we take 534 minus 507 to get 27. 4

5 Finally, we need to get our sum of squares within or S/A. Notice that we have already calculated the information that we need for this formula in the previous steps. We take to get 20. Insert this information into your ANOVA summary table. Now divide by its df to change it to variance. Variance is called Mean Square. Once you have done this, you are ready to find F. F = MS between/ MS within which is 27/2 = Always write results statistically. F (1, 10) = 13.50, p <.05 Note that F critical = I found this in my F table. Our interpretation is that the groups are significantly different. The treatment had an effect on the dependent variable. The two groups likely did not come from the same population. So, how do we know if F-obtained is significant? It s just like a one-tailed z-test or a one-tailed t-test! If Fobs > Fcrit, then you have a significant effect. If Fobs < Fcrit, then you do not have a significant effect. Your textbook also provides information on how to calculate an ANOVA using Excel as well as the other statistical tests. Interpretation is very important with any statistical test that you calculate. The F value alone only tells us about the significant of the test. We also want to know how the groups differ. Let s return to the therapy example from earlier. Type of therapy significantly affects symptoms of Simple Phobia. Or, there is a main effect of type of therapy on Simple Phobia. In addition to interpretation, you should also write the results statistically. Here is an outline of how to do that. F (dfbetween, dfwithin) = Fobs, p <.05 We place F in front to signify that this is an F test, then we place the dfb and dfwithin in paranthesis. The F-obtained value follows the equal sign. The p <.05 tells us that the test is significant. If the F is not significant, we would use p >.05. All statistical analyses have assumptions associated with them. For ANOVA, the two primary assumptions are: 1.The population from which the samples were taken are normally distributed 2. The samples are drawn from a population of equal variances. This is also called homogeneity of variance ANOVA, like t-test, is robust. The F-test is minimally effected by violations of normality and relatively insensitive to violations of homogeneity of variance. Thus, if the data is not normal or the variances are not equal, you can still calculate the tests, but should still be cautious in your interpretation. Just because a test is significant that does not tell us that the results are also important. Effect size tells us if the effect is important. In other words, it tells us how much variance 5

6 in the dependent variable can be explained by the independent variable. Recall that an effect size may be small, medium, or large. The larger the effect the greater the importance. Here I have presented one formula for calculating effect size. This is the formula for eta squared. In order to calculate eta squared, you need to divide the SS between by the SS total. This is fairly easy to do given that you calculate the effect size after you have calculated the entire ANOVA. Just get the numbers that you need from your ANOVA summary table. Power is another important concept. Power is the ability to detect an effect if there is one. Power is influenced by several factors such as sample size, alpha, effect size, and variance. A larger sample size is related to more power. Also, larger the effect size is related to more power. Smaller variance is related to more power as well. Finally, setting alpha at.05 is related to more power than setting alpha at.01. Think about critical values for.05 and.01 respectively. The cut-offs for.01 are more stringent than for.05 making it more difficult to detect an effect. 6