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Like the chi‐square test, the t‐test provides a significance value called a p‐value, and is presented the same way. Correlation Coefficients What are correlation coefficients? Correlation coefficients measure the strength of association between two variables, and reveal whether the correlation is negative or positive. A negative relationship means that when one variable
increases the other decreases (e.g., drinking alcohol and reaction time). A positive relationship means that when one variable increases so does the other (e.g.,study time and test scores). Correlation scores range from ‐1 (strong negative correlation) to 1 (strong positive correlation). The closer the figure is to zero, the weaker the association, regardless whether it is a negative or positive integer.
Table 6. Example of chi‐square.
A chi‐square statistic was then performed to determine if type of library worked at affected whether librarians had heard the term evidence‐based practice. As you can see by the table below, p>.05, therefore there is no statistical difference in distribution of awareness of EBP based on the type of library worked at.
Value Df Sig. Chi‐Square 16.955 4 .990
Why use a chi‐square? A chi‐square is the statistic being used here because the relationship between two ordinal variables (type of library worked at and awareness of the term EBP) is being explored. What does value mean? It is simply the mathematical calculation of the chi‐square. It is used to then derive the p‐value, or significance. What does df mean? Df stands for degrees of freedom. Degrees of freedom is the number of values that can vary in the estimation of a parameter. It is calculated for the chi‐square statistic by looking at the cross tabulation and multiplying the number of rows minus one by the number of columns minus one (r‐ 1) x (c‐1). In this case, if we look back to Fig. 4, we can see that we have a two by five table. Thus, (2‐ 1) x (5‐1) = 4. What does sig. mean? Sig. stands for significance level, or p‐level. In this case p = .990. As this number is larger than .05, the null hypothesis is proven. There is no statistically relationship between type of library and awareness of EBP, despite the differences in percentages we saw in Table 5.
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When should you use correlation coefficients? Correlation coefficients should be used whenever you want to test the strength of a relationship. There are many tests to measure correlation; which one to use depends on what variables you are examining. A few are listed below: Nominal variables: Phi, Cramer’s V, Lambda, Goodman and Kruskal’s Tau Ordinal variables: Gamma, Sommers D, Spearman’s Rho Ratio variables: Pearson r Limitations of correlation coefficients Correlation does not indicate causality. Simply because there is a relationship between two variables does not mean that one causes the other. Keep in mind correlation only looks at the relationship between two variables; there many be others affecting the relationship (remember the confounding variable!). Correlation
coefficients can also be skewed by outlier values. How do I know if the relationship is statistically significant? Correlation scores range from ‐1 (strong negative correlation) to 1 (strong positive correlation). The closer the figure is to zero, the weaker the association, regardless of whether it is a negative or positive integer. Analysis of Variance (ANOVA) What is ANOVA? Like the t‐test, ANOVA compares means, but can be used to compare more than two groups. ANOVA looks at the differences between categories to see if they are larger or smaller than those within categories. When should you use ANOVA? The dependent variable in ANOVA must be ratio. The independent variable can be
Table 7. Example of a t‐test.
An independent samples t‐test was performed to determine if there was a statistical difference between genders on the Evidence‐based Practice test. As the table below illustrates, there was a significant difference in performance between males and females, t (19)=‐.398 p<.05
Value df Sig. T‐test ‐.398 19 .049
Why use a t‐test? A t‐test is used for these variables because we are comparing the mean of one variable (EPB Test Score, a ratio variable) between 2 groups (sex, a nominal variable). An independent samples t‐test is used here because the groups being compared are mutually exclusive ‐ male and female. How is the t‐test interpreted? The t‐test value, degrees of freedom, and significance values can be interpreted in precisely the same way as the chi‐square in Fig. 5. The significance value of .049 is less that .05, therefore it can be stated that the null hypothesis is disproved; there is a statistical significant difference between the performance of male librarians and the performance of female librarians on the EBP Perceptions Test.
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Table 8. Example of a Pearson r correlation. nominal or ordinal, but most be composed of mutually exclusive groups Limitations of ANOVA ANOVA measures whether there are significant differences between three or more groups, but it does not illustrate where the significance lies – there could be differences between all groups or only two. There are tests called post hoc comparisons which can be performed to determine where significance lies, however. How do I know if the relationship is statistically significant? An ANOVA uses an f‐test to determine if there is a difference between the means of groups. The f‐test can be used to calculate a p‐score, which is analyzed in the same way as chi‐squares and t‐tests. Statistical Significance and Effect Size Measures Significance tests have a couple of weaknesses. One is the fairly arbitrary value at which statistical significance is said to have occurred. Why is α = .051 not a significant finding while α = .049 is? The
second disadvantage is that significance tests do not give an indication of the strength of a relationship, merely that it exists. A smaller significance value could be the result of a larger sample rather than a strong relationship. This is where effect sizes come in. Effect sizes are tests which gauge the strength of a relationship. There are many different effect size indices; which to use depends on the statistical test being performed. Multivariate Analysis Any in‐depth discussion of multivariate analysis is beyond the scope of a paper entitled “Statistical Primer”; however, here is a brief introduction. Multivariate analysis looks at the relationship between more than two variables, for example length of service and type of librarian might together be predictors of perception of EBP. Using bivariate statistical methods, it is not possible to see the relationship between two independent variables as well as their effect on the dependent variable. There are several multivariate statistical methods. Here are two of the most common.
A Pearson r correlation was performed to determine if there was a relationship between age and score on the EBP test instrument. The correlation revealed that the two were significantly related, r=+.638, n=210, p<.05. Why was a Pearson r correlation performed? A Pearson r was done because both variables involved, Age and EBP Perceptions Test score, are ratio variables. What does the r value tell us? The r is correlation score. Remember that correlation scores range from +/‐1 to 0. Therefore, a score of +.638 reveals that there is a strong positive correlation between age and EBP score. The fact that it is positive means that when one variable increases so does the other – the older the librarian, the higher they scored on the EBP test instrument.
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Table 9. Example of ANOVA. Statistical Test Effect Size Measure Comments
Chi‐square phi Phi tests return a value between zero (no
relationship) and one (perfect relationship).
T‐test Cohen’s d
Cohen’s d results are interpreted as 0.2 being a small effect, 0.5 a medium and 0.8 a large effect size. (Cohen 157)
ANOVA Eta squared Eta square values range between zero and one, and can be interpreted like phi and Cohen’s d.
Table 10. Statistical tests and effect size measures. Multivariate analysis of variance (MANOVA) is an ANOVA which analyses several dependent variables. It can be interpreted in much the same way as ANOVA tests. MANOVA has advantages over doing multiple ANOVA tests, including reducing the potential for Type I errors (concluding that there is a relationship when there is not). Conversely, MANOVA tests can also reveal relationships not apparent in ANOVA tests. Multiple linear regression examines “the relationship between one ‘effect’ variable,
called the dependent or outcome variable, and one or more predictors, also called independent variables” (Muijs 168). It is designed to work with continuous variables, though there are different techniques available for analyzing other variable types. While performing and analyzing regressions are complicated, they are valuable tools for examining the relationship between many variables. It is important to note that, like other inferential statistical techniques, values are created that provide the statistical significance of the relationships.
For the EBP Test Instrument Score, the analysis of variance (ANOVA) revealed that there was not a significant difference in performance F (3, 47)=3.43, p<.05 between types of librarians. The critical value (.245) for the scores was obtained the F distribution table using dfbetween=4 and dfwithin=16. Why was an ANOVA performed? An ANOVA was the appropriate statistical technique because the dependent variable (EBP Test score) is continuous, while the independent variable (type of library worked at) is nominal and composed of several groups. What does this tell us? The F test score was calculated at 3.43. This score was used in conjunction with the degrees of freedom (because we are comparing several groups, there are two degrees of freedom scores, one for between the groups (4) and one for within the groups (16) to calculate the p‐score. P = .245, which is greater than .05. Therefore there is no difference in performance on the test based on the type of library worked at.
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Conclusion This paper is not intended to produce statistical experts. Rather, it is a guide to understanding the basic principles and techniques common in library and related research. Most statistical software packages, such as SPSS or SAS, will effortlessly perform statistics, so it is far more important that as a researcher you know a) how to select an appropriate sample; b) know what statistical technique is appropriate in which situations; and c) be able to interpret results correctly. There are a few things you can do to make yourself more comfortable with statistics. One is to purchase a basic quantitative methods textbook. Look for one that comes with a CD of sample data sets. Running through the exercises in the textbook will provide you with valuable practice in performing and analyzing statistics. There are several textbooks available in the library field, although any social science quantitative methods texts
would be useful. The second thing you can do is to read the research literature in your field. If you know the topic well, it is easier to evaluate and interpret results. Works cited Cohen, J. “A Power Primer.” Psychological
Bulletin 112 (1992): 155‐159. Muijs, Daniel. Doing Quantitative Research in
Education with SPSS. London: SAGE, 2004.
Nardi, Peter M. Doing Survey Research: A
Guide to Quantitative Methods. Boston: Allyn and Bacon, 2003.
Weisberg, Herbert F., Jon A. Krosnick, and
Bruce D. Bowen. An Introduction to Survey Research, Polling, and Data Analysis. 3rd ed. Thousand Oaks, Calif.: Sage Publications, 1996.
