Abstract
In the previous chapter, we discussed the basic principles of inferential statistics and emphasized the difference between descriptive and inferential statistics. The latter allows the researcher to make inferences about the population based on the observations collected. Through analytical tools, inferential statistics help researchers decide whether the difference between groups is statistically significant enough to support our hypothesis that the difference exists in the population.
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Notes
- 1.
See Banerjee et al. (2009).
- 2.
*Note that this is the hypothesized difference between the population means, which we denoted in Step 2 as the null hypothesis.
- 3.
Whether paired measures or repeated measures, individual scores are ultimately matched together in order to appropriately utilize the formula. For this reason, the dependent sample t test is also referred to as the matched t test.
- 4.
Note that the sample size (n) is not the number of total observations, rather the number of matched observations.
- 5.
Though we will be focusing only on one-way ANOVA, there exist other types of ANOVA, such as two-way, covariate, and multivariate (see Chiappelli 2014). A special case of ANOVA occurs during a repeated measure design, in which a single group is repeatedly measured multiple times against different treatments. The utility of ANOVA for repeated measures is similar to the related discussion under dependent samples t test.
- 6.
See McHugh (2011).
- 7.
The formula for its calculation can be both tedious and time-consuming but was essential for statisticians and investigators in the nineteenth and twentieth centuries. Lucky for us, the computing of r today is most often done (easily) by any good statistical software program. The formula is provided here for continuity purposes; however, its calculation by hand will not be further discussed.
- 8.
An important clarification is made, particularly for regression discussed in the next section.
- 9.
The correlation coefficient is unit-less because when the terms are placed in the actual formula, the units in the numerator and denominator cancel each other out.
- 10.
Do not let the word “positive” in positive association lead you to believe that this relationship is exclusive to variables that increase together. A positive relationship may also be used to describe two variables that decrease together. What is more important to understand is that they behave in a similar manner.
- 11.
Although this may be a useful heuristic, having the actual correlation coefficient provides a much more accurate and precise description.
- 12.
See Furr (n.d.).
- 13.
Recall from correlations that this linear line was simply imagined—however, here, we actually graph this line.
- 14.
This add-on also makes the equation akin to the generalizability (G) theory, briefly described in Chap. 3. The G theory is a statistical framework that aims to fractionate the error in the measurements we make, which ultimately allows our findings to be closer and closer to the true value: X = T + ε.
- 15.
The same is true for β, the parametric regression coefficient. However, β is standardized via z-transformation in order to be able to describe the parameter; for this reason, b is denoted as the unstandardized regression coefficient and β as the standardized regression coefficient. See Chiappelli (2014) and Bewick et al. (2003).
Bibliography
Chiappelli F. Comparing two groups: T tests family (13,14,15) [PowerPoint slides]; n.d.
Recommended Reading
Banerjee A, Chitnis UB, Jadhav SL, Bhawalkar JS, Chaudhury S. Hypothesis testing, type I and type II errors. Ind Psychiatry J. 2009;18(2):127–31. https://doi.org/10.4103/0972-6748.62274.
Chiappelli F. Fundamentals of evidence-based health care and translational science. Heidelberg: Springer; 2014.
McHugh ML. Multiple comparison analysis testing in ANOVA. Biochem Med (Zagreb). 2011;21(3):203–9.
Furr RM. Testing the statistical significance of a correlation. Winston-Salem, NC: Wake Forrest University; n.d.
Bewick V, Cheek L, Ball J. Statistics review 7: correlation and regression. Crit Care. 2003; 7(6), 451–459. Print.
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Khakshooy, A.M., Chiappelli, F. (2018). Inferential Statistics II. In: Practical Biostatistics in Translational Healthcare. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57437-9_6
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