Learning Objectives
After reading this chapter you should understand:
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The logic of hypothesis testing.
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The steps involved in hypothesis testing.
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What test statistics are.
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Types of error in hypothesis testing.
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Common types of t-tests, one-way and two-way ANOVA.
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How to interpret SPSS outputs.
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Notes
- 1.
Note that the power of a statistical test may depend on a number of factors which may be particular to a specific testing situation. However, power nearly always depends on (1) the chosen significance level, and (2) the magnitude of the effect of interest in the population.
- 2.
Note that most tests follow the same scheme. First, we compare a sample statistic, such as the sample mean, with some standard or the mean value of a different sample. Second, we divide this difference by another measure (i.e., the standard deviation or standard error), which captures the degree of uncertainty in the sample data.
- 3.
To obtain the critical value, you can also use the TINV function provided in Microsoft Excel, whose general form is “TINV(α, df).” Here, α represents the desired Type I error rate and df the degrees of freedom. To carry out this computation, open a new Excel spreadsheet and type in “=TINV(2*0.05,9).” Note that we have to specify “2*0.05” (or, directly 0.1) under α as we are applying a one-tailed instead of a two-tailed test.
- 4.
Unfortunately, there is quite some confusion about the difference between α and p-value. See Hubbard and Bayarri (2003) for a discussion.
- 5.
We don’t have to conduct manual calculations and tables when working with SPSS. However, we can easily compute the p-value ourselves using the TDIST function in Microsoft Excel. The function has the general form “TDIST(t, df, tails)”, where t describes the test value, df the degrees of freedom and tails specifies whether it’s a one-tailed test (tails = 1) or two-tailed test (tails = 2). For our example, just open a new spreadsheet and type in “=TDIST(2.274,9,1)”. Likewise, there are several webpages with Java-based modules (e.g., http://www.graphpad.com/quickcalcs/index.cfm) that calculate p-values and test statistic values.
- 6.
The number of pairwise comparisons is calculated as follows : k·(k − 1)/2, with k the number of groups to compare.
- 7.
Field (2013) provides a detailed introduction to further ANOVA types such as multiple ANOVA (MANOVA) or an analysis of covariance (ANCOVA).
- 8.
Note that you can also apply ANOVA when comparing two groups, but as this will lead to the same results as the independent samples t-test, the latter is preferred.
- 9.
Nonparametric alternatives to ANOVA are, for example, the χ2-test of independence (for nominal variables) and the Kruskal–Wallis test (for ordinal variables). See, for example, Field (2013).
- 10.
In fact, these two assumptions are interrelated, since unequal group sample sizes result in a greater probability that we will violate the homogeneity assumption.
- 11.
SS is an abbreviation of “sum of squares” because the variation is calculated by means of squared differences between different types of values.
- 12.
Note that the application of post hoc tests only makes sense when the overall F-test finds a significant effect.
- 13.
Note that the p-value is not truly zero as SPSS just truncates the rightmost digits. If you double-click on the output and again on the p-value, you will see that the factual p-value is 0.000217.
- 14.
Note. however, that you SPSS offers the option to compute η 2 by going to Analyze ► Compare Means ► Means. Under Options, you can request an ANOVA table including η 2.
References
Boneau, C. A. (1960). The effects of violations of assumptions underlying the t test. Psychological Bulletin, 57(1), 49–64.
Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364–367.
Cho, H. C., & Abe, S. (2012). Is two-tailed testing for directional research hypotheses tests legitimate? Journal of Business Research, 66(9), 1261–1266.
Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159.
Field, A. (2013). Discovering statistics using SPSS (4th ed.). London: Sage.
Hubbard, R., & Bayarri, M. J. (2003). Confusion over measure of evidence (p’s) versus errors (α’s) in classical statistical testing. The American Statistician, 57(3), 171–178.
Lilliefors, H. W. (1967). On the Kolmogorov–Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62(318), 399–402.
Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38(3/4), 330–336.
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Sarstedt, M., Mooi, E. (2014). Hypothesis Testing & ANOVA. In: A Concise Guide to Market Research. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-53965-7_6
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