Discussion Questions: Applied Statistics

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 What is a Type I error? Explain how the cumulative Type I error affects your decision making. How are the two independent sample t-tests different from ANOVA?

During Statistical Hypothesis Testing, incorrect conclusions can be made. Type 1 error occurs when there is incorrect rejection of a null hypothesis when it is in fact true.

Cumulative Type 1 error affects statistical data. It is given by 1-0.99^n where n is the number of times. The cumulative type 1 error increases with n. carrying several t-tests results to deciding the statistical significance of the means based on probability. Multiple t tests increases chances of performing type 1 error hence one-way ANOVA is applied.

t test satisfactorily determines significance difference for two sample means and is inappropriate where more than two sample means are compared and ANOVA is instead used.

Analysis of Variance (ANOVA) is independent and allows extrapolation of results of t-tests. ANOVA generalizes t-test to several groups. It is a statistical test determining whether group’s means are equal. Two independent t-tests would give a higher probability of incurring type 1 error. ANOVA is advantageous than performing two t-tests and is essential for comparison of more than three means. It is a statistical test to determine if several groups’ means are equal hence generalizing t-test in several groups. To test if the group means under one dependent variable differ significantly, one-way ANOVA is applied (Robert, 2004).Why is the F distribution important? How do you determine if a significant difference exists among the groups in ANOVA? How do you determine differences between the groups in ANOVA?

F distribution is frequent null distribution in ANOVA. F test assumes normally distributed population means with similar standard deviation are equal. It assesses if there is a difference in the expected values of groups. Multiple comparisons are possible. A single test detects the possible differences. F ratio is based on a single F distribution tail in the ANOVA. ANOVA is not a one-tailed test. F ratio is given as a function of squared differences between sample mean and grand mean. F-test compares the total deviation. It is a test of significance in comparisons of two population variance estimates. If ANOVA involves five groups estimates of variance are tested for heterogeneity before conducting the ANOVA.

If there are equal cases number in a sample Hartley’s Maximum F ratio is used where maximum F ratio is calculated from the data by division of smallest variance estimate with the largest estimate. For example if variation between sample means is larger than variation within sample means large calculated F value results, hence null hypothesis is rejected thus, there is a difference between the population means. F value indicates average difference between the group means relative to each group average variance. Post-hoc statistical test is conducted to examine if each group mean significantly differ from other group mean after getting the overall F value (Henry, 2005).

Describe the requirements that must be met before an ANOVA test may be used. Discuss what the researcher must do if one of these requirements is not met.

One way ANOVA require independent observations thus makes the statistical analysis easier. It assumes that samples are normally distributed and have equal variance. Homogeneity in variance of group data is crucial. It also assumes that any observation is a result of total of several components. Unequal sample variances correlated with sample means indicate treatment effects as well as sample error component is not additive. In such a case, a treatment may be multiplied by error. A researcher could use models to determine the change in values of response variables. This is the Fixed-effects model which applies several treatments and assists the researcher to have estimates of response variables given by a treatment in a population. Random effects model are applicable in case of variation in treatment especially when sampling several factor levels (random variables) in a great population. Conducting a priori contrast where mean of a group is compared to other combined groups (Timothy, 2005).

Reference

Henry E. Klugh. (2005) Statistics: The Essentials for Research. New York: Wiley Publishers.

Robert A. Donnelly. (2004). The Complete Idiot’s Guide to Statistics.USA: Penguin Group, Inc.

Timothy C. Urdan. (2005). Statistics in Plain English. New Jersey: Lawrence Erlbaum Associates, Inc.