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# Houser ppt chapt16

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### Houser ppt chapt16

1. 1. Chapter 16 Analysis and Reporting of Quantitative Data
2. 2. Inferential Analysis <ul><li>Enables inference of results from a carefully selected sample to an overall population </li></ul><ul><li>Quantifies the potential effects of error on the results </li></ul>
3. 3. General Rules for Quantitative Analysis <ul><li>Statistical tests are selected a priori </li></ul><ul><ul><li>The acceptable level of significance is also selected before analysis begins </li></ul></ul><ul><li>Run all of the identified tests </li></ul><ul><li>Report all of the tests that were run </li></ul><ul><ul><li>Selective reporting is a source of bias </li></ul></ul>
4. 4. Type of Analysis Driven By… <ul><li>The goals of the analysis </li></ul><ul><li>Assumptions of the data </li></ul><ul><li>Number of variables in the analysis </li></ul>
5. 5. Statistical Inference <ul><li>Intended to answer two fundamental questions: </li></ul><ul><ul><li>How probable is it that the differences between observed results and those expected on the basis of the null hypothesis have been produced by chance alone? </li></ul></ul><ul><ul><li>How reliable are the results obtained? </li></ul></ul>
6. 6. Statistical Inference <ul><li>Questions about reliability answered by setting confidence limits </li></ul><ul><li>Questions about probability answered by hypothesis testing </li></ul><ul><li>Conclusions concerned with probability of drawing an erroneous conclusion </li></ul>
7. 7. Estimation <ul><li>Point estimate : this sample statistic equals the population parameter </li></ul><ul><li>Interval estimate : the range of numbers we believe will include the population parameter </li></ul><ul><li>Statistical estimation allows for determination of the amount of uncertainty in the estimate </li></ul>
8. 8. Confidence Interval Two numerical values defining an interval that we believe, with an identified level of confidence, actually includes the estimated population parameter
9. 9. Calculation of Confidence Interval for the Mean <ul><li>Determination of an acceptable confidence level a priori (1-  ) </li></ul><ul><li>Identification of coefficient Z  </li></ul><ul><li>Calculation of standard error (s /  n) </li></ul><ul><li>Determination of the mean </li></ul>
10. 10. Hypothesis Testing <ul><li>Statistical test of significance </li></ul><ul><ul><li>Between a sample and a known population </li></ul></ul><ul><ul><li>Between two samples </li></ul></ul><ul><ul><li>Between two variables in a sample </li></ul></ul><ul><li>Can be used to test differences between: </li></ul><ul><ul><li>Means </li></ul></ul><ul><ul><li>Proportions </li></ul></ul><ul><ul><li>Variances </li></ul></ul>
11. 11. Inferential Statistics <ul><li>If a difference is detected, may be due to: </li></ul><ul><ul><li>Experimental treatment caused the effect </li></ul></ul><ul><ul><li>Sampling error caused the effect (chance) </li></ul></ul><ul><li>We cannot prove the experiment caused the difference </li></ul><ul><li>We can estimate the probability it was caused by error </li></ul>
12. 12. Steps in Hypothesis Testing <ul><li>1. State research question as a statistical hypothesis </li></ul><ul><li>2. Select the level of significance for the statistical test (alpha) </li></ul><ul><li>3. Decide on the appropriate test statistic </li></ul><ul><li>4. Determine the value the test statistic must attain to be declared significant (critical value) </li></ul><ul><li>5. Perform the calculations for the test statistic </li></ul><ul><li>6. Apply decision rule and draw conclusions </li></ul>
13. 13. Hypotheses <ul><li>H 0 : The null hypothesis </li></ul><ul><ul><li>e.g. H 0 :  = 30 </li></ul></ul><ul><li>H A : The alternative hypothesis </li></ul><ul><ul><li>e.g. H A :  30 </li></ul></ul><ul><li>The null hypothesis is rejected only if data presents sufficiently strong evidence to support the alternative </li></ul>
14. 14. Steps in Hypothesis Testing <ul><li>1. State research question as a statistical hypothesis </li></ul><ul><li>2. Select the level of significance for the statistical test (alpha) </li></ul><ul><li>3. Decide on the appropriate test statistic </li></ul><ul><li>4. Determine the value the test statistic must attain to be declared significant (critical value) </li></ul><ul><li>5. Perform the calculations for the test statistic </li></ul><ul><li>6. Apply decision rule and draw conclusions </li></ul>
15. 15. Level of Significance <ul><li>Level of Significance: </li></ul><ul><ul><li>Preset standard that is considered significant in determining differences </li></ul></ul><ul><ul><li>Compare to p value: the probability of a type I error that the researcher is willing to risk </li></ul></ul><ul><ul><li>Most common: .05 and .01 </li></ul></ul><ul><li>Set a priori (alpha) </li></ul>
16. 16. Steps in Hypothesis Testing <ul><li>1. State research question as a statistical hypothesis </li></ul><ul><li>2. Select the level of significance for the statistical test (alpha) </li></ul><ul><li>3. Decide on the appropriate test statistic </li></ul><ul><li>4. Determine the value the test statistic must attain to be declared significant (critical value) </li></ul><ul><li>5. Perform the calculations for the test statistic </li></ul><ul><li>6. Apply decision rule and draw conclusions </li></ul>
17. 17. Most Inferential Tests... <ul><li>Based upon difference between parameter estimates in the sample and the population or between two samples </li></ul><ul><li>Difference between n and N estimates </li></ul><ul><li>Standard error </li></ul>
18. 18. Steps in Hypothesis Testing <ul><li>1. State research question as a statistical hypothesis </li></ul><ul><li>2. Select the level of significance for the statistical test (alpha) </li></ul><ul><li>3. Decide on the appropriate test statistic </li></ul><ul><li>4. Determine the value the test statistic must attain to be declared significant (critical value) </li></ul><ul><li>5. Perform the calculations for the test statistic </li></ul><ul><li>6. Apply decision rule and draw conclusions </li></ul>
19. 19. Distributions for Probability <ul><li>Each parameter has its own distribution </li></ul><ul><ul><li>Variance: F </li></ul></ul><ul><ul><li>Proportion:  2 </li></ul></ul><ul><ul><li>Mean: t </li></ul></ul><ul><ul><li>Correlation: t </li></ul></ul><ul><li>Each finite sample will have its own unique probability distribution </li></ul><ul><li>Closer to normality as sample size increases </li></ul>
20. 20. Steps in Hypothesis Testing <ul><li>1. State research question as a statistical hypothesis </li></ul><ul><li>2. Select the level of significance for the statistical test (alpha) </li></ul><ul><li>3. Decide on the appropriate test statistic </li></ul><ul><li>4. Determine the value the test statistic must attain to be declared significant (critical value) </li></ul><ul><li>5. Perform the calculations for the test statistic </li></ul><ul><li>6. Apply decision rule and draw conclusions </li></ul>
21. 21. Hypothesis test of means <ul><li>One sample tests of means </li></ul><ul><li>H 0 : x =  </li></ul><ul><li>H A : x  </li></ul><ul><li>Calculate test statistic: t = x -  </li></ul><ul><li> SE (x) </li></ul><ul><li>Compare to critical value from t distribution </li></ul><ul><li>Apply decision rule to reject / do not reject the null hypothesis </li></ul>
22. 22. Steps in Hypothesis Testing <ul><li>1. State research question as a statistical hypothesis </li></ul><ul><li>2. Select the level of significance for the statistical test (alpha) </li></ul><ul><li>3. Decide on the appropriate test statistic </li></ul><ul><li>4. Determine the value the test statistic must attain to be declared significant (critical value) </li></ul><ul><li>5. Perform the calculations for the test statistic </li></ul><ul><li>6. Apply decision rule and draw conclusions </li></ul>
23. 23. Decision Rules <ul><li>All possible values of the test statistic are divided into 2 regions </li></ul><ul><li>Rejection region </li></ul><ul><ul><li>Those values of the test statistic with a low probability of occurrence if H 0 is true </li></ul></ul><ul><ul><li>Statistically constructed at 5% </li></ul></ul><ul><ul><li>Critical value: cut point for the rejection region </li></ul></ul>
24. 24. Possible Decisions <ul><li>A significant difference is detected between the actual values and the hypothesized values: </li></ul><ul><ul><li>Reject the H 0 </li></ul></ul><ul><li>A significant difference is not detected between the actual values and the hypothesized values </li></ul><ul><ul><li>Do not reject the H A </li></ul></ul>
25. 25. Type I and II Errors
26. 26. Type I and II Errors <ul><li>Type I Error: Treatment doesn’t work, but we think it does </li></ul><ul><ul><li>Leads to unwarranted change </li></ul></ul><ul><ul><li>Control by appropriately setting  </li></ul></ul><ul><li>Type II Error: Treatment works, but we think it doesn’t </li></ul><ul><ul><li>Leads to missed opportunities </li></ul></ul><ul><ul><li>Designated as  , controlled by sampling </li></ul></ul>
27. 27. The t test of means <ul><li>Differences between mean values: </li></ul><ul><ul><li>Between a sample and a known population value </li></ul></ul><ul><ul><ul><li>One sample t </li></ul></ul></ul><ul><ul><li>Between two independent samples </li></ul></ul><ul><ul><ul><li>Independent samples t test </li></ul></ul></ul><ul><ul><li>Between two time periods for the same group </li></ul></ul><ul><ul><ul><li>Paired samples t test </li></ul></ul></ul>
28. 28. The Chi Square test <ul><li>Tests for differences in rates, proportions, or probabilities </li></ul><ul><ul><li>Between a sample proportion and a known population proportion </li></ul></ul><ul><ul><ul><li>Chi square test of model fit </li></ul></ul></ul><ul><ul><li>Between two independent samples </li></ul></ul><ul><ul><ul><li>Chi square test of independence </li></ul></ul></ul><ul><ul><li>Between two variables in a single sample </li></ul></ul><ul><ul><ul><li>Chi square test of association </li></ul></ul></ul>
29. 29. Analysis of Variance <ul><li>Tests for differences in means in more than two groups </li></ul><ul><ul><li>Single dependent variable </li></ul></ul><ul><ul><ul><li>ANOVA </li></ul></ul></ul><ul><ul><li>Single dependent variable with potential covariates </li></ul></ul><ul><ul><ul><li>ANCOVA </li></ul></ul></ul><ul><ul><li>Single dependent variable measured over more than two time periods </li></ul></ul><ul><ul><ul><li>Repeated Measures ANOVA </li></ul></ul></ul>
30. 30. Most Common Reported Statistics <ul><li>Descriptive statistics about sample and variables </li></ul><ul><li>Analysis of group equivalency </li></ul><ul><li>Statistics about the role of error </li></ul><ul><li>Statistics to evaluate magnitude of effect </li></ul><ul><li>Statistics to determine confidence level </li></ul>