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CHAPTER 16: ANALYSIS OF VARIANCE (ONE WAY-Factor)

Key Terms

Analysis of variance (ANOVA) – An overall test of the null hypothesis fro more than two population means.

One-way ANOVA – The simplest type of analysis of variance that tests whether differences exist among population means categorized by only one factor or independent variable.

Treatment effect – The existence of at least one difference between the population means categorized by the independent variable.

Variability between groups – Variability among scores of subjects who, being in different groups, receive different experimental treatments.

Variability within groups – variability among scores of subjects who, being in the same group, receive the same experimental treatment.

Random error – The combined effects (on the score of individual subjects) of all uncontrolled factors

Sum of squares (SS) – The sum of the squared deviations of some set of scores about their mean

Degrees of freedom (df) – The number of deviations free to vary in any sum of square term.

Mean Square (MS) – A variance estimate obtained by dividing a sum of squares by its degrees of freedom.

F-ratio – Ratio of the between-group mean squares (for subjects treated differently) to the within-group mean square (for subjects treated similarly.)

Squared curvilinear correlation – The proportion of variance in the dependent variable that can be explained by the independent variable.

Multiple comparisons – the series of possible comparisons whenever more than two population means are involved.

Scheffe’s test – A multiple comparison test that, regardless of the number of comparisons, never permits the cumulative probability of at least one type I error to exceed the specified level of significance.

Text Review

            Testing the null hypothesis for more than two population means requires a statistical procedure known as (1)______________. Specifically, Chapter 22 deals with (2) ______________. ANOVA, where population means differ with respect to only one factor. One source of variability in ANOVA is the differences between group means. Small differences can be attributed to (3) ______________. However, relatively large difference between group means probably indicate that the null hypothesis is (4) _____________. If there is at least one difference between the population means, there is (5) _____________. A second source of variability in ANOVA is an estimate of the variability within groups (subjects treated similarly). To make a decision about the null hypothesis, these two sources of variability are compared. The more that the variability between groups exceeds the variability within groups, the more likely the null hypothesis will be false. Regardless of whether the null hypothesis is true or false, the variability within groups reflects only (6)_____________. Random error is the combined effects of all uncontrolled factors such as individual differences among subjects, variations in experimental conditions, and measurement errors. The within-group variability estimate is often referred to as the (7) _____________.

            For three or more samples, the null hypothesis is tested with the F ratio, variability between groups divided by variability within groups. F has its own family of sampling distributions, so an F table must be consulted to find the critical F value. In the F test, if the variability between groups sufficiently exceeds the variability within groups, then the null hypothesis will be rejected. If the null hypothesis is true, then the two estimates of variability (Between and within groups) will reflect only random error. The values will be similar, so the F will be small and the null hypothesis will be retained.

            Variance is a measure of (8) _____________. A variance estimate indicates that information from a sample is used to determine the unknown variance for a population. In ANOVA, a variance estimate is composed of the numerator, the sum of squares, and the denominator, which are always (9) _____________. When the ratio is calculated, it produces the mean of the squared deviations referred to as (10) _____________.

            In ANOVA, most of the computational efforts is in finding the various sum of squared terms. SSbetween equals the sum of the squared deviations of group means about their mean, the overall mean.

SSwithin equals the sum of the squared deviations of all scores about their respective group means. SStotal equalsthe sum of the squared deviations of all scores about the overall mean. Calculations of the sum of squares terms can be verified by calculating all three from scratch and then checking because the sum of squares total equals the sum of squares within and the sum of squares between added together. For each sum of squares term, degrees of freedom differ. In ANOVA, the degrees of freedom for sum of squares total always equals the combined degrees of freedom for the other sum of squares terms.

            Mean squaresbetween reflects the variability between groups who are treated (11) _____________. Mean squareswithinreflects the variability among scores for subjects who are treated (12) ___________ Mean squareswithin measures only (13)_____________, but mean squaresbetween measures (14) _____________. The observed F, once calculated, may be compared with the critical F specified by the pair of degrees of freedom associated with it. Rejection of the null hypothesis indicates only that not all population means are equal.

            The assumptions for the F test are the same as for t. All underlying populations are assumed to be (15)_____________ with equal variances violations of the assumptions are not critical as long as sample size is greater than (16) ___. ANOVA techniques used in the text presume that scores are independent. Furthermore, attention should be paid to sample size so that it is not unduly small or excessively large. Whenever a statistical significant F has been obtained, the researcher should

                                                                                  2

consider using the squared curvilinear correlation, n (n squared), and Cohen’s rule of thumb to estimate (17)_____________independently of sample size.

            In order to pinpoint the one or more differences between pairs of population means that contribute to the rejection, a test of (18) ___________________ must be used. Multiple t tests cannot be used because it would increase the probability of a (19)               error. Once the overall null hypothesis has been rejected in ANOVA, Scheffe’s test can be used for all possible comparisons without the probability of the type I error exceeding the (20)                                      . When sample sizes are unequal, Scheffe’s critical value must be calculated for each comparison. But when sample sizes are equal, the critical mean difference is only calculated once and then used to evaluate the remaining comparisons. It is important to note that Scheff’s test should be used only when the overall null hypothesis has been rejected.

            The F test in ANOVA is equivalent to a (21) _____________ test even though the rejection region appears only in the upper tail of the distribution. This is due to the squaring of all the values that makes it impossible to have a negative value for F.

Answer 1

Testing the null hypothesis for more than two population means requires a statistical procedure known as (1) Analysis of variance(ANOVA). Specifically, Chapter 22 deals with (2) one way ANOVA. ANOVA, where population means differ with respect to only one factor. One source of variability in ANOVA is the differences between group means. Small differences can be attributed to (3) null. However, relatively large difference between group means probably indicate that the null hypothesis is (4) rejected. If there is at least one difference between the population means, there is (5) treatment effect. A second source of variability in ANOVA is an estimate of the variability within groups (subjects treated similarly). To make a decision about the null hypothesis, these two sources of variability are compared. The more that the variability between groups exceeds the variability within groups, the more likely the null hypothesis will be false. Regardless of whether the null hypothesis is true or false, the variability within groups reflects only (6) variablity between groups. Random error is the combined effects of all uncontrolled factors such as individual differences among subjects, variations in experimental conditions, and measurement errors. The within-group variability estimate is often referred to as the (7) variablity within group.

            For three or more samples, the null hypothesis is tested with the F ratio, variability between groups divided by variability within groups. F has its own family of sampling distributions, so an F table must be consulted to find the critical F value. In the F test, if the variability between groups sufficiently exceeds the variability within groups, then the null hypothesis will be rejected. If the null hypothesis is true, then the two estimates of variability (Between and within groups) will reflect only random error. The values will be similar, so the F will be small and the null hypothesis will be retained.

            Variance is a measure of (8) variablity . A variance estimate indicates that information from a sample is used to determine the unknown variance for a population. In ANOVA, a variance estimate is composed of the numerator, the sum of squares, and the denominator, which are always (9) degrees of freedom. When the ratio is calculated, it produces the mean of the squared deviations referred to as (10) mean square.

            In ANOVA, most of the computational efforts is in finding the various sum of squared terms. SSbetween equals the sum of the squared deviations of group means about their mean, the overall mean.

SSwithin equals the sum of the squared deviations of all scores about their respective group means. SStotal equalsthe sum of the squared deviations of all scores about the overall mean. Calculations of the sum of squares terms can be verified by calculating all three from scratch and then checking because the sum of squares total equals the sum of squares within and the sum of squares between added together. For each sum of squares term, degrees of freedom differ. In ANOVA, the degrees of freedom for sum of squares total always equals the combined degrees of freedom for the other sum of squares terms.

            Mean squaresbetween reflects the variability between groups who are treated (11) between groups. Mean squareswithinreflects the variability among scores for subjects who are treated (12) within subjects Mean squareswithin measures only (13) within group mean square, but mean squaresbetween measures (14) between group mean square. The observed F, once calculated, may be compared with the critical F specified by the pair of degrees of freedom associated with it. Rejection of the null hypothesis indicates only that not all population means are equal.

            The assumptions for the F test are the same as for t. All underlying populations are assumed to be (15) independent with equal variances violations of the assumptions are not critical as long as sample size is greater than (16) 3 or more. ANOVA techniques used in the text presume that scores are independent. Furthermore, attention should be paid to sample size so that it is not unduly small or excessively large. Whenever a statistical significant F has been obtained, the researcher should

                                                                                  2

consider using the squared curvilinear correlation, n (n squared), and Cohen’s rule of thumb to estimate (17) proportion of variance independently of sample size.

            In order to pinpoint the one or more differences between pairs of population means that contribute to the rejection, a test of (18) multiple comparison must be used. Multiple t tests cannot be used because it would increase the probability of a (19) random error. Once the overall null hypothesis has been rejected in ANOVA, Scheffe’s test can be used for all possible comparisons without the probability of the type I error exceeding the (20) specified level of significance . When sample sizes are unequal, Scheffe’s critical value must be calculated for each comparison. But when sample sizes are equal, the critical mean difference is only calculated once and then used to evaluate the remaining comparisons. It is important to note that Scheff’s test should be used only when the overall null hypothesis has been rejected.

            The F test in ANOVA is equivalent to a (21) t test even though the rejection region appears only in the upper tail of the distribution. This is due to the squaring of all the values that makes it impossible to have a negative value for F.