Inferential Statistics

1. Inferential Statistics Statistical Analysis of Research Data

2. Statistical Inference • Getting information about a population from a sample • How practical are “statistically significant” results? • Cost/benefit • Crucial difference • Client acceptability • Public and political acceptability • Ethical and legal concerns

3. Inferential Statistics • These provide a means for drawing conclusions about a population given the data actually obtained from the sample. They allow a researcher to make generalizations to a larger number of individuals, based on information from a limited number of subjects. They are based on: • Probability theory • Statistical inference • Sampling distributions

4. Inferential Statistics • Probability theory – the basis for decision-making statistical inferences. It refers to a large number of experiences, events or outcomes that will happen in a population in the long run. Likelihood and chance are similar terms. Examples are usually based on tossing a coin and finding heads or tails. Probabilities are statements of likelihood expressed in values from 0 to 1.0. • p = the number of outcomes the total possible outcomes

5. Inferential Statistics • Statistical inference – statistics enable us to judge the probability that our inferences or estimates are close to the truth • Sampling distributions – are theoretical distributions developed by mathematicians to organize statistical outcomes from various sample sizes so that we can determine the probability of something happening by chance in the population from which the sample was drawn. They allow us to know the relative frequency or probability of occurrence of any value in the distribution.

6. Inferential Statistics • Hypothesis testing – 5 basic steps • Make a prediction • Decide on a statistical test to use • Select a significance level and a critical region (region of rejection of the null hypothesis). To do this you must consider two things • Whether both ends (tails) of the distribution should be included. • How the critical region of a certain size will contribute to Type I or Type II errors.

7. Critical Region in the Sampling Distribution for a One-Tailed Test

8. Critical Regions in the Sampling Distribution for a Two-Tailed Test

9. Levels of Significance • Remember, if a printsout shows a two-tailed test result, and you wanted a one-tailed result, divide the two tailed p value by 2. • Example: p = .080 (two-tailed) or p>.05 • p = .040 (one-tailed) or p<.05 • The first would not be statistically significant, whereas, the second would be statistically significant

10. Outcomes of Statistical Decision Making

11. Inferential Statistics • Hypothesis testing cont. • Computing the test statistic - The test statistic is not a mean, sd or any form of descriptive data. It is simply a number that can be compared with a set of results predicted by the sampling distribution • Compare the test statistic to the sampling distribution (table) and make a decision about the null hypothesis – reject it if the statistic falls in the region of rejection. • Consider the power of the test – its probability of detecting a significant difference – parametric tests are more powerful

13. Degrees of Freedom • One sample t-test or paired t-test = N-1 • Independent t-test = N-2 • Chi-square test = (# rows - 1) x (# columns – 1) ANOVA : df between groups = (# levels or groups – 1) df within groups = (# subjects - # of levels) Correlations = N-2

14. Levels of Measurement • There are four levels or scales of measurement, Each level is classified according to certain characteristics. Data that fall in the first level are limited to certain statistical tests. Choices of statistical tests (and the power of the tests) increase as the levels go up.

15. Levels of Measurement cont. • Nominal scale – measurement at its weakest – numbers or other symbols are used to classify or partition a class into mutually exclusive subclasses – animals can be classified as dogs, cats, etc. You can test hypotheses regarding distribution among the categories by using the Chi-square test.

16. Levels of Measurement cont. • Ordinal scale – shows relationships among classes, such as higher than, more difficult than, etc. It allows the attributes of a variable to be ranked in relation to each other. A researcher can test hypotheses using non-parametric statistics of order and ranking.

17. Levels of Measurement cont. • Interval scale – is similar to the ordinal scale, but the distance between any two numbers is of a known size. The numbers used have absolute values and the interval between each number is considered to be equal. Increasing amounts of a variable are represented by increasing numbers on the scale. The variable is continuous. There is no true zero – where you have none of the variable. All parametric tests can be used with interval data.

18. Levels of Measurement cont. • Ratio scale – is like the interval scale but it has a true zero point as its origin. Time, length and weight are ratio scales when used alone, but not as a characteristic of a person. Arithmetic, all parametric tests and geometric means can be used with ratio data.

19. Tests of Significance • Parametric tests of significance – used if there are at least 30 observations, the population can be assumed to be normally distributed, variables are at least in an interval scale • Z tests are used with samples over 30. There are four kinds (two samples or two categories) • t-tests are used when samples are 30 or less. • Single sample t-test (one sample) • Independent t-test (two samples) • Paired t-test (two categories

20. Tests of Significance • Non-parametric tests of significance – small numbers, can’t assume a normal distribution, or measurement not interval • Chi-square – requires only nominal data – allows researcher to determine whether frequencies that have been obtained in research differ from those that would have been expected – use a X2 sampling distribution • Chi-square goodness of Fit • Chi-Square test of independence

21. Tests of Significance • Mann Whitney U – an alternate to the independent t-test – must have at least ordinal data. It counts the comparative ranks of scores in two samples (from highest to lowest) The null hypothesis is that the two samples are randomly distributed. Use U sampling distribution tables for small sample sizes (1-8) and medium sample sizes (9-20) and the Z test for large samples

23. Tests of Significance • Wilcoxin Matched Pairs (signed rank test) – is an alternate to the paired t-test. It is used for repeated measures on the same individual. It requires a measurement between ordinal and interval scales – the scores must have some real meaning. Use a T table. If the T is less than or equal to the T in the table, the null hypothesis is rejected.

24. Measures of Association • Parametric Measures of Association – These answer the question, “within a given population, is there a relationship between one variable and another variable?” A measure of association can exist only if data can be logically paired. It can be tested for significance. • Correlation – answers the question, “What is the degree of relationship between “x” and “y” – Use Pearson Product Moment Correlation (Pearson r ) –see next slide

25. Measures of Association The Pearson Correlation Coefficient (Pearson r) • The r examines the relationship between two quantitative sets of scores. • The r varies from – 1.00 to + 1.00 • The r is not a proportion and cannot be multiplied by 100 to get a percentage. • To think of the r as a percentage, it needs to be converted to the “Coefficient of Determination” or R2 . An r of .50 is 25% better than an r of 0.00

26. Measures of Association • Non-parametric tests for association • Correlation • The Spearman Rank Order Correlation (Rs)– “To what extent and how strongly are two variables related?” • Phi coefficient – it can be used with nominal data, but should have ordinal data • Kendall’s Q – can be used with nominal data

27. Prediction • Parametric Prediction – using a correlation, if you know score “x”, you can predict score “y” for one person – Use regression analysis • Simple linear regression – allows the prediction from one variable to another – you must have at least interval level data • Multiple linear regression – this allows the prediction of one variable from several other variables. The dependent variable must be on the interval scale

28. Prediction • Non-parametric Prediction – measures the extent to which you can reduce the error in predicting the dependent variable as a consequence of having some knowledge of the independent variable such as, predicting income [DV] by education [IV] • Kendall’s Tau – used with ordinal data and ranking - is better than the Gamma because it takes ties into account • Gamma - used with ordinal data to predict the rank of one variable by knowing rank on another variable • Lambda – can be used with nominal data – knowledge of the IV allows one to make a better prediction of the DV than if you had no knowledge at all

29. Parametric Multiple Comparisons • The analysis of variance (ANOVA) is probably the most commonly encountered multiple comparison test. It compares observed values with expected values in trying to discover whether the means of several populations are equal. It compares two estimates of the population variance. One estimate is based on variance within each sample – within groups. The other is based on variation across samples – between groups. The between group variance is the explained variance (due to the treatment) and the variation within each group is the unexplained variance (the error variance).

30. Parametric Multiple Comparisons • ANOVA cont. The ratio of the explained scores to the unexplained scores gives the F statistic. If the variance between the groups is larger, giving an F ratio greater than 1, it may be significant depending upon the degrees of freedom. If the F ratio is approximately 1, it means that the null hypothesis is supported and there was no significant difference between the groups.

31. Parametric Multiple Comparisons • ANOVA cont. If the null hypothesis is rejected, then one would be interested in determining which groups showed a significant difference. The best way to check this is to conduct a post hoc test such as the Tukey, Bonferrioni, or Scheffe. (SPSS will do this for you if you click on Post-hoc and check the test desired.Check on descriptives while you still in ANOVA, and the program will also give you the mean for each group)

32. Parametric Multiple Comparisons • Two-Way Analysis of Variance • Classifies participants in two-ways • Results answer three questions • Two main effects • An interaction effect

33. Non-parametric Multiple Comparison • Kruskal-Wallis Test – an alternative to the one-way ANOVA. The scores are ranked and the analyses compare the mean rank in each group. It determines if there is a difference between groups. • McNemar Test – an adaptation of the Chi-square that is used with repeated measures at the nominal level. • Friedman Test –an alternative to the repeated ANOVA. Two or more measurements are taken from the same subjects. It answers the questions as to whether the measurement changes over time.