Applied Univariate, Bivariate, and Multivariate Statistics. Daniel J. Denis

Second, did it change its shape at all? Why or why not?(c) Compute the covariance between parent height and child height. Does the sign of the covariance suggest a positive or negative relationship?(d) Standardize the covariance by computing Pearson r. Interpret the obtained correlation coefficient, and test it for statistical significance using either SPSS or R.

      52 2.52. Consider the following data on whether a student passed or failed a mathematics course (grade = 0 is “failed” and grade = 1 is “passed”), along with that student's study time for the course, in average minutes per day for the duration of the course: grade studytime 0 30 0 25 0 59 0 42 0 31 1 140 1 90 1 95 1 170 1 120Conduct an independent‐samples t‐test on this data using SPSS and R. Verify that the assumption of homogeneity of variances is met in SPSS.

      53 2.53. A researcher is interested in conducting a two‐sample t‐test between a treatment group and a control group. The researcher anticipates an effect size of approximately d = 1.5 and wishes to test the null hypothesis μ1 = μ2 at a significance level of 0.05. Estimate required sample size assuming the researcher wishes to attain power of at least 0.90 for her test of the null hypothesis.

Further Discussion and Activities

      1 2.54. As discussed in this chapter, null hypothesis significance testing (NHST) has been critically evaluated and dissected as a means for drawing scientific inferences in the social and natural sciences. Rozeboom (1960) quite nicely summarized the main criticisms in The Fallacy of the Null‐Hypothesis Significance Test. Read the article and discuss Rozeboom's distinction between decisions versus degrees of belief. Why is such a distinction important for a scientist to understand the difference between statistical versus scientific inference? Rozeboom's article can be downloaded from Christopher D. Green's Classics in the History of Psychology website: http://psychclassics.yorku.ca/Rozeboom/

      2 R.A. Fisher, the modern “father of statistics” wrote in 1956:“… no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas.”Many writers and researchers, however, have found that since the inception of the significance test in the early 1900s, scientists, both social and otherwise, routinely employ the 0.05 level of significance in rejecting null hypotheses. Read Mindless Statistics by Gigerenzer (2004), and discuss the dangers and risks, both practical and theoretical, of allowing the “null ritual” to dominate in science.

      Notes

      1 ¹Classics in the History of Psychology is an on‐line educational resource hosted by Christopher D. Green of York University in Toronto, Canada. It contains a huge selection of milestone papers and articles in the history of psychology. It can be accessed via http://psychclassics.yorku.ca/.

      2 ² We can also extend the binomial distribution to one in which instead of n trials giving rise to r occurrences, we have n trials giving rise to outcomes in k categories:where x is now a vector of random variables x = [x1, x2, …, xk]′.

      3 ³ For a more technical demonstration of how and why this convergence occurs, see Proschan (2008).

      4 ⁴ For details on the gamma function, see Degroot and Schervish (2002, p. 295). A plot of the gamma function appears as follows (see Crawley, 2013, p. 264, for the R code):

      5 ⁵ Power will be discussed later in this chapter.

      6 ⁶ Though in this text we define consistency of an estimator quite simply, further distinctions exist between weak and strong consistency. See Shao (2003, pp. 132–133).

      7 ⁷ We can also distinguish between weaker vs. stronger forms of the theorem. For details, see Casella & Berger (2002, pp. 236–238).

      8 ⁸ For an overview of alternative correlation coefficients such as the biserial, point-biserial and tetrachoric coefficients, see Howell (2002) or Warner (2013).

      9 ⁹ The coefficient appears in Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15, 72–101.

      10 ¹⁰ G*Power is a user-friendly statistical program that can be downloaded for free at: https://www.psychologie.hhu.de/arbeitsgruppen/allgemeine‐psychologie‐und‐arbeitspsychologie/gpower.html.

      11 ¹¹> r <- c(0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.99) > r_squared <- r^2 > r_squared [1] 0.0100 0.0400 0.0900 0.1600 0.2500 0.3600 0.4900 0.6400 0.8100 0.9801 > d <- sqrt((4*r^2)/(1-r^2)) > d [1] 0.2010076 0.4082483 0.6289709 0.8728716 1.1547005 1.5000000 [7] 1.9603921 2.6666667 4.1294832 14.0358479

      12 ¹² Melatonin is sometimes used as a non‐prescription sleep aid.