1. Numeracy & Quantitative Methods: Level 7 – Advanced Quantitative Analysis

2. Choice of which test to use depends on: number of samples (k) if 2 or more samples, whether these are independent or dependent data type (parametric and non parametric) This slide series: two or more samples parametric data (independent sample t-test, dependent sample t-test, analysis of variance) Choosing statistical tests

3. Purpose To test whether the two groups are drawn from the same population and share the same characteristics – i.e. distribution and mean. In your research, if you’d looked at age and gender and calculated the mean weekly income for both, the next stage is to find out if the observed difference is statistically significant. Independent samples t-test is a calculation based on the mean, standard deviation and sample size for both groups

4. 1. The dependent variable is interval or ratio in measurement 2. Random sampling technique used to collect data from the population 3. Independence of observation 4. Normally distribution 5. Homogeneity of variance: the distribution of the two groups is the same (Levene is not significant)

5. H 0 : no difference in the mean between the two groups. H 1 : there is a difference between the two groups Example: H 0 = no difference between the mean weekly income of men and women Dependent variable = age Independent variable = gender.

6. Levene test for homogeneity of variance (also referred to as the F-test of sample variance) H 0 : the two groups are drawn from the same population and share the same variance/distribution Look at the descriptive statistics generated to see what the standard deviations of the two groups are

8. Determine the null hypothesis for t-test H o : the two groups are drawn from the same population and share the same mean Look at means of two groups Depending on the outcome of the Levene test look at either the ‘equal variances assumed t test’ or the ‘equal variances not assumed t test’

10. 1. Same distribution & same mean 2. Same distribution & different mean 3. Different distribution & same mean 4. Different distribution & different mean

15. State null hypotheses and purpose of tests Report results for Levene and standard deviations Select correct t-test Report significances and means Discuss whether the differences are what you would expect

16. Test data are collected from the same participants at two points in time (i.e. are dependent samples) Repeated measure – so appropriate for experimental research design Example: measuring someone’s IQ before tutoring and then again IQ level after tutoring. H0: no difference in IQ level pre and post tutoring H1: there is a difference in IQ level Dependent samples t-test output interpreted in the same way as independent but without Levene’s.

17. When working with three or more samples (bus users, train users and pedestrians) need to take a slightly different approach Analysis of Variance or ANOVA compares the mean scores of three or more groups of data.

18. Example: It is thought that literacy level is a function of gender and the length of time spent in education. Does gender and the length of time spent in education affect on literacy levels? As with t-testing, we need to compare the sample means. But, even if all the population means were identical, the sample means are unlikely to be exactly equal — there will be always be some differences due to sampling variation.

19. Question really is: “Are the observed differences between the sample means simply due to sampling variation or due to real differences in the population means?” So, cannot just look at the sample means — we also need to look at the variability of what is being measured. Analysis of variance compares the variability between the groups (how far apart are the means?) to the variability within the groups (how much natural variation is there in our measurements?).

20. ANOVA measures two sources of variation – the way groups differ internally versus what the difference is between them – and compared their relative sizes. variation BETWEEN groups: for each data value look at the difference between its group mean and the overall mean variation WITHIN groups: for each data value we look at the difference between that value and the mean of its group

21. Having separated the total variation these separate sources of variation are then tested with an F-Test. The ANOVA F-statistic is a ratio of the Between Group variation divided by the Within Group variation: If the F ratio is less than 0.05 then the differences between the conditions studied is regarded as statistically significant. A large F is evidence against H 0 – indicates that there is more difference between groups than within groups.

22. All ANOVAs are parametric ANOVAs determine if the mean differences between conditions are significantly different from each other 1.Observations are normally distributed 2.Homogeneity of variance – conditions or treatments should not affect the variability between observations 3.Sphericity – equal covariance of the error measures and is required in “within subjects” design. Note: for assumptions to stand the appropriate test needs to be p>.05.

23. Example: whether the IQ of respondents depends on which of the 4 types of tutoring they have received. IQ = factor (independent treatment variable) Different tutoring = levels In the example, there is only one factor, IQ, so our analysis of the effect of tutoring on IQ is a one-way ANOVA. Adding in another factor, say length of tutoring would make it a two-way ANOVA...and so on...

24. One-way ANOVA H 0 : there is no difference in the population means of the different levels of factor A (the only factor) H 1 : the means are not the same. Two-way ANOVA H o : (1) no difference in the means of factor A (2) no difference in means of factor B (3) no interaction between factors A and B H 1 : for (1) & (2) the means are not equal. For (3) there is an interaction between A and B.

25. The ANOVA has told us that there is a significant difference between at least two of the three conditions, and therefore we can reject our null hypothesis that the mean of the three conditions are the same (Ho: μ1=μ2=μ3). But, we don’t know if there is a significant difference between the control condition and either or both of the two other conditions, or if there is a significant difference between the two conditions. Visual inspection of the samples will tell us something but in most cases a formal follow up or post hoc analysis is necessary.

26. Least Significant Difference (LSD) (t-test): when number of comparisons of interest is small or when there are at most 3 conditions to be compared. Scheffe: when there are 4 or more conditions (in between subjects) AND either unequal numbers of subjects per conditions OR there are serious doubts about the assumptions of normality and/or equal variances. Bonferroni: when there are more than 4 conditions in within subjects designs. Tukey ‟ s Honestly significant Test: when there are 4 or more conditions (in between subjects) AND equal numbers of subjects per condition AND there are no known problems with the assumptions.

27. David, M. and Sutton, C. (2011) Social Research : An Introduction. 2nd ed. London: Sage. Fielding, J. and Gilbert, N. (2006) Understanding social statistics. 2 nd ed. London: Sage. References

28. This resource was created by the University of Plymouth, Learning from WOeRk project. This project is funded by HEFCE as part of the HEA/JISC OER release programme.Learning from WOeRk This resource is licensed under the terms of the Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales license (http://creativecommons.org/licenses/by-nc-sa/2.0/uk/).http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ The resource, where specified below, contains other 3 rd party materials under their own licenses. The licenses and attributions are outlined below: 1.The name of the University of Plymouth and its logos are unregistered trade marks of the University. The University reserves all rights to these items beyond their inclusion in these CC resources. 2.The JISC logo, the and the logo of the Higher Education Academy are licensed under the terms of the Creative Commons Attribution -non-commercial-No Derivative Works 2.0 UK England & Wales license. All reproductions must comply with the terms of that license. Author Laura Lake InstituteUniversity of Plymouth Title Advanced Quantitative Analysis Description Bivariate Analysis: Parametric Date Created July 2011 Educational Level Postgraduate (Level 7) Keywords Parametric, independent and dependent t-tests, analysis of variance, UKOER, LFWOER, CPD, Learning from WOeRK, UOPCPDRM, Continuous professional development, HEA, JISC, HEFCE Back page originally developed by the OER phase 1 C-Change project ©University of Plymouth, 2010, some rights reserved