Oneway ANOVA comparing 3 or more means

Overall Purpose A Oneway ANOVA is used to compare three or more average scores. A Oneway ANOVA is used to compare three or more average scores. Used when there is one IV with 3 or more levels and one DV. Used when there is one IV with 3 or more levels and one DV. Sample data are used to answer a question about population means. Sample data are used to answer a question about population means.

Examples Does a new instructional method lead to more favorable student outcomes when compared to two types of traditional instruction comparison groups? Does a new instructional method lead to more favorable student outcomes when compared to two types of traditional instruction comparison groups?

Examples Do low-income preschool children who live in the following situations differ in school readiness? Do low-income preschool children who live in the following situations differ in school readiness? – father present in the home – see father regularly – no contact with their father

The F statistic The F statistic is a ratio. The F statistic is a ratio. The denominator is within group variance. The denominator is within group variance. The numerator is between group The numerator is between group + within group variance. + within group variance. t 2 =F for the two group case. t 2 =F for the two group case.

The F statistic If the IV has no relationship to the population means, the F statistic will equal 1. If the IV has no relationship to the population means, the F statistic will equal 1. If F=1, there is no between group variance. If F=1, there is no between group variance.

The F statistic The F statistic examines whether sample means are varying more than they would be expected to vary due to sampling error alone. The F statistic examines whether sample means are varying more than they would be expected to vary due to sampling error alone.

Assumptions Normality Normality Homogeneity of Variance Homogeneity of Variance Independence of Observations Independence of Observations Random Sampling Random Sampling

Statistical Significance How do you know when there is a statistically significant difference between the average scores you are comparing? How do you know when there is a statistically significant difference between the average scores you are comparing?

Statistical Significance When the F statistic is greater than 1 by enough to be beyond sampling error. When the F statistic is greater than 1 by enough to be beyond sampling error. We know this because the p value is less than alpha, usually set at.05. We know this because the p value is less than alpha, usually set at.05.

Statistical Significance A small p value tells us that there is a low probability that the variability in the means is due to sampling error alone. A small p value tells us that there is a low probability that the variability in the means is due to sampling error alone. We conclude there is more variability between the group means than would be expected by sampling error alone. We conclude there is more variability between the group means than would be expected by sampling error alone.

Hypotheses Hypotheses for the Oneway ANOVA: Hypotheses for the Oneway ANOVA: Null Hypothesis:  1 =  2  3...  k  1 =  2  3...  k Alternative Hypothesis:  i =/=  j for at least one pair.  i =/=  j for at least one pair. At least two of the population means are different. At least two of the population means are different.Where: k = the number of population means k = the number of population means

Additional Considerations There is a unique critical F value for each degrees of freedom condition. There is a unique critical F value for each degrees of freedom condition. A statistically significant F statistic does not tell us where the difference lies. A statistically significant F statistic does not tell us where the difference lies. Confidence intervals and effect sizes can be very helpful in interpreting the results. Confidence intervals and effect sizes can be very helpful in interpreting the results.

Example Our research design: Our research design:

The Research Question Are classroom structural characteristic (class size, number of ELL children, etc.) different across the three stress groups? Are classroom structural characteristic (class size, number of ELL children, etc.) different across the three stress groups?

Writing About Results Use APA format for reporting test statistics and p values: Use APA format for reporting test statistics and p values: – t(29) = 7.345, p=.005 – F(1,123) = 2.446, p =.122 Recognize the distinction between a statistically significant finding and an important finding. Recognize the distinction between a statistically significant finding and an important finding.

Writing About Results Remember to review the writing guidelines in the handout on the website. Remember to review the writing guidelines in the handout on the website.