# Best Practice: How to Write a Dissertation or Thesis Quantitative Chapter 4

In the first paragraph of your quantitative chapter 4, the results chapter, restate the research questions that will be examined. This reminds the reader of what you’re going to investigate after having been trough the details of your methodology. It’s helpful too that the reader knows what the variables are that are going to be analyzed.

Spend a paragraph telling the reader how you’re going to clean the data. Did you remove univariate or multivariate outlier? How are you going to treat missing data? What is your final sample size?

The next paragraph should describe the sample using demographics and research variables. Provide frequencies and percentages for nominal and ordinal level variables and means and standard deviations for the scale level variables. You can provide this information in figures and tables.

Here’s a sample:

**Frequencies and Percentages.** The most frequently observed category of Cardio was Yes (*n* = 41, 72%). The most frequently observed category of Shock was No (*n* = 34, 60%). Frequencies and percentages are presented in Table 1.

**Summary Statistics.** The observations for MiniCog had an average of 25.49 (*SD* = 14.01, *SE** _{M}* = 1.87, Min = 2.00, Max = 55.00). The observations for Digital had an average of 29.12 (

*SD*= 10.03,

*SE*

*= 1.33, Min = 15.50, Max = 48.50). Skewness and kurtosis were also calculated in Table 2. When the skewness is greater than 2 in absolute value, the variable is considered to be asymmetrical about its mean. When the kurtosis is greater than or equal to 3, then the variable’s distribution is markedly different than a normal distribution in its tendency to produce outliers (Westfall & Henning, 2013).*

_{M}Now that the data is clean and descriptives have been conducted, turn to conducting the statistics and assumptions of those statistics for research question 1. Provide the assumptions first, then the results of the statistics. Have a clear accept or reject of the hypothesis statement if you have one. Here’s an independent samples t-test example:

**Introduction.** An two-tailed independent samples *t*-test was conducted to examine whether the mean of MiniCog was significantly different between the No and Yes categories of Cardio.

**Assumptions.** The assumptions of normality and homogeneity of variance were assessed.

** Normality.** A Shapiro-Wilk test was conducted to determine whether MiniCog could have been produced by a normal distribution (Razali & Wah, 2011). The results of the Shapiro-Wilk test were significant,

*W*= 0.94,

*p*= .007. These results suggest that MiniCog is unlikely to have been produced by a normal distribution; thus normality cannot be assumed. However, the mean of any random variable will be approximately normally distributed as sample size increases according to the Central Limit Theorem (CLT). Therefore, with a sufficiently large sample size (

*n*> 50), deviations from normality will have little effect on the results (Stevens, 2009). An alternative way to test the assumption of normality was utilized by plotting the quantiles of the model residuals against the quantiles of a Chi-square distribution, also called a Q-Q scatterplot (DeCarlo, 1997). For the assumption of normality to be met, the quantiles of the residuals must not strongly deviate from the theoretical quantiles. Strong deviations could indicate that the parameter estimates are unreliable. Figure 1 presents a Q-Q scatterplot of MiniCog.

** Homogeneity of variance.** Levene’s test for equality of variance was used to assess whether the homogeneity of variance assumption was met (Levene, 1960). The homogeneity of variance assumption requires the variance of the dependent variable be approximately equal in each group. The result of Levene’s test was significant,

*F*(1, 54) = 18.30,

*p*< .001, indicating that the assumption of homogeneity of variance was violated. Consequently, the results may not be reliable or generalizable. Since equal variances cannot be assumed, Welch’s t-test was used instead of the Student’s t-test, which is more reliable when the two samples have unequal variances and unequal sample sizes (Ruxton, 2006).

**Results.** The result of the two-tailed independent samples *t*-test was significant, *t*(46.88) = -4.81, *p* < .001, indicating the null hypothesis can be rejected. This finding suggests the mean of MiniCog was significantly different between the No and Yes categories of Cardio. The mean of MiniCog in the No category of Cardio was significantly lower than the mean of MiniCog in the Yes category. Table 3 presents the results of the two-tailed independent samples *t*-test. Figure 2 presents the mean of MiniCog(No) and MiniCog(Yes).

In the next paragraphs, conduct stats and assumptions for your other research questions. Again, assumptions first, then the results of the statistics with appropriate tables and figures.

Be sure to add all of the in-text citations to your reference section. Here is a sample of references.

**References**

Conover, W. J., & Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. *The American Statistician, 35*(3), 124-129.

DeCarlo, L. T. (1997). On the meaning and use of kurtosis. *Psychological Methods,* 2(3), 292-307.

Levene, H. (1960). Contributions to Probability and Statistics. *Essays in honor of Harold Hotelling,* I. Olkin et al. eds., Stanford University Press, 278-292.

Razali, N. M., & Wah, Y. B. (2011). Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. *Journal of Statistical Modeling and Analytics, 2*(1), 21-33.

Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student’s t-test and the Mann-Whitney U test. *Behavioral Ecology, 17*(4), 688-690.

Intellectus Statistics [Online computer software]. (2019). Retrieved from https://analyze.intellectusstatistics.com/

Stevens, J. P. (2009). *Applied multivariate statistics for the social sciences* (5th ed.). Mahwah, NJ: Routledge Academic.

Westfall, P. H., & Henning, K. S. S. (2013). *Texts in statistical science: Understanding advanced statistical methods.* Boca Raton, FL: Taylor & Francis.