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Analyzing experimental research data

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Researchers use several tools and procedures for analyzing quantitative data obtained from different types of experimental designs. Different designs call for different methods of analysis. This presentation focuses on:
T-test
Analysis of variance (F-test), and
Chi-square test

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Analyzing experimental research data

  1. 1. Analyzing Experimental Research Data: The T-test ANOVA and Chi-square Atula Ahuja Changyan Shi
  2. 2.  The experimental design determines the statistical test to be used to analyze the data.  There are several tools and procedures for analyzing quantitative data obtained from different types of experimental designs. Different designs call for different methods of analysis. This presentation focuses on: 1. T-test 2. Analysis of variance (F-test), and 3. Chi-square test Experimental Design and Statistics
  3. 3. The Logic of Significance Testing  The results of an inferential statistical test informs whether the results of an experiment would occur frequently or rarely by chance.  Inferential statistical test with small p values occur frequently by chance (accept the null hypothesis), whereas large values occur rarely by chance (reject the null hypothesis).
  4. 4. The Logic of Significance Testing  P value is the probability at which the null hypothesis will be rejected when it is true.  Traditionally statisticians say that any event that occurs by chance 5 times or fewer in 100 occasions is a rare event. (i.e., .05 level of significance).
  5. 5. T- Test  When the means of two independent groups are to be compared, we can use the T- test. This test can help determine how confident we can be that the differences between two groups as a result of the treatment is not due to chance. The researcher calculates a t-value using the sample mean and standard deviation and compares the calculated t-value against a tabulated value. If null hypothesis is rejected, we can say that the difference between the two groups is significant.
  6. 6. Example for t-test A researcher compares performances of two randomly selected groups learning French. The two groups, follow up their frontal lessons with practice sessions.  the Experimental Group gets practice sessions with the aid of the computer.  the Control group has practice sessions with a teacher. The researcher investigates the effects of the computer practice session on students’ achievement in French.
  7. 7. Hypotheses testing 210 μμ:H  211 μμ:H  Mean scores of the two groups are equal 2 ˆˆ μˆ-μˆ 2 1 1 21 nn tcal    Suppose, upon calculation, the researcher finds the tcal = 1.99 The researcher can use t-test to test the hypothesis
  8. 8. Example for t-test  Upon comparison with the table value of T at p=0.05, it is found that tcal > ttab  This means the null hypothesis is rejected and the differences between two groups are significantly different.  The result is reported as t=1.99, p=.05
  9. 9. One-Way ANOVA When there are more than two groups, the appropriate procedure is ‘ANOVA’ where we need to analyze the variability within groups and variability between groups. The test we do in ANOVA is the F-test
  10. 10. One-Way ANOVA  3210 μμμ:H samethearemeanspopulationtheofallNot:1H The “F-test” groupswithinyVariabilit groupsbetweenyVariabilit F 
  11. 11. Example TM1 TM2 TM3 TM4 60 50 40 57 67 52 45 67 42 43 43 54 67 67 55 67 56 67 46 69 62 59 61 69 64 67 45 68 59 64 52 65 72 63 53 70 71 65 63 68
  12. 12. Example Step 1) calculate the sum of squares between groups: Mean for group 1 = 62.0 Mean for group 2 = 59.7 Mean for group 3 = 50.3 Mean for group 4 = 65.4 Grand mean= 59.85 SSB = [(62-59.85)2 + (59.7-59.85)2 + (50.3-59.85)2 + (65.4-59.85)2 ] x n per group= 19.65x10 = 1266.6 , n= number of observations in each group TM1 TM2 TM3 TM4 60 50 40 57 67 52 45 67 42 43 43 54 67 67 55 67 56 67 46 69 62 59 61 69 64 67 45 68 59 64 52 65 72 63 53 70 71 65 63 68
  13. 13. Example Step 2) calculate the sum of squares within groups: (60-62) 2+(67-62) 2+ (42-62) 2+ (67-62) 2+ (56-62) 2+ (62- 62) 2+ (64-62) 2+ (59-62) 2+ (72-62) 2+ (71-62) 2+ (50- 59.7) 2+ (52-59.7) 2+ (43- 59.7) 2+67-59.7) 2+ (67- 59.7) 2+ (69-59.7) 2…+…… = 2060.6 Mean=62 Mean=59.7 Mean=50.3 Mean=65.4 TM1 TM2 TM3 TM4 60 50 40 57 67 52 45 67 42 43 43 54 67 67 55 67 56 67 46 69 62 59 61 69 64 67 45 68 59 64 52 65 72 63 53 70 71 65 63 68
  14. 14. Step 3) Fill in the ANOVA table 3 1266.6 422.2 7.38 0.001 36 2060.6 57.2 Source of variation d.f. Sum of squares Mean Sum of Squares F-statistic p-value Between Within Total 39 3327.2 F value= 𝑉𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑔𝑟𝑜𝑢𝑝/𝑑𝑓 𝑉𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑤𝑖𝑡ℎ𝑖𝑛 𝑔𝑟𝑜𝑢𝑝/𝑑𝑓
  15. 15. Factorial ANOVA (Two way)  When two factors or more than two factors are involved.(age, gender. level of competence)  The aim is to test if there is also an interaction between teaching method and gender/age/etc.
  16. 16. Factorial ANOVA (Two way) Gender Teaching method 1 Teaching method 2 Teaching method 3 Teaching method 4 1 60 50 48 47 1 67 52 49 67 1 42 43 50 54 1 67 67 55 67 1 56 67 56 68 2 62 59 61 65 2 64 67 61 65 2 59 64 60 56 2 72 63 59 60 2 71 65 64 65
  17. 17. Factorial ANOVA (Two way) TM 1 TM 2 TM 3 TM 4 Average Male Mean=58.4 N=5 Mean=55.8 N=5 Mean=51.6 N=5 Mean=60.6 N=5 Mean=56.6 N=20 Female Mean=65.6 N=5 Mean=63.6 N=5 Mean=61 N=5 Mean=62.2 N=5 Mean=63.1 N=20 Average 62 N=10 59.7 N=10 56.3 N=10 61.4 N=10 59.9 N=40 1μˆ1μˆ
  18. 18.  Using the between, within and interaction sum of squares, we create the ANOVA table and calculate F- Statistic associated with main effects and interaction effects  Hypothesis test procedure will be the same as before
  19. 19. Chi Square (Χ2) Where o = observed frequencies, and e = expected frequencies  The most obvious difference between the chi-square tests and the other hypothesis tests we have considered (t and ANOVA) is the nature of the data.  For chi-square, the data are frequencies rather than numerical scores.  Chi Squared is used to observe the difference between what we actually observe and what we expect to find if the null hypothesis is true.  The chi-square statistic is calculated as follows
  20. 20. Data was collected on citizen’s viewpoints about building of the 2012 Olympic venue at Stratford and tried to find if viewpoints changed according to the perspectives of different groups. Through a survey/questionnaire, 20 responses from each category of local person were collected, about the usefulness of the new Olympic developments. The statement posed: ‘The 2012 Olympic Games development will be of benefit to the whole community of Stratford, east London.’ 1 2 3 4 Strongly agree Agree Disagree Strongly disagree Case Study: Using chi-squared to analyse questionnaire responses
  21. 21. Results of the survey. Category (type) Frequency of negative responses (Observed values: o) Business owner 4 School student 6 Adult male resident 14 Adult female resident 10 Senior citizen 16 20 people responded from each category and only the frequency of negative response, i.e. those who either disagreed or strongly disagreed with the statement.
  22. 22. The expected data (e) is the mean negative frequency of response, calculated by adding up the observed data (o) and then dividing by the number of categories, i.e. 5. This gives an expected frequency of 10 for each category. Business owner School student Adult male resident Adult female resident Senior citizen Total o 4 6 14 10 16 50 e 10 10 10 10 10 50 o - e -6 -4 4 0 6 -------- (o – e)² 36 16 16 0 36 -------- (o – e)² e 3.6 1.6 1.6 0 3.6 -------- x² 3.6 1.6 1.6 0 3.6 10.4
  23. 23. Interpreting the Chi-Squared Value Calculated chi-square value= 10.4 4 degrees of freedom. Critical values for 4 df are: Confidence level 0.10 90% 0.05 95% 0.01 99% 0.005 99.5% Critical value 7.78 9.49 13.28 14.86 To reject the null hypothesis (Hₒ), chi-squared score must be greater than the critical value at the 0.05 level of significance. Since 10.4 is higher than the 0.05 level of significance- which is 9.49, we can reject the null hypothesis (Hₒ).
  24. 24. Summary of different statistical procedures Different types of data analysis are appropriate for different types of research problems.  Qualitative: data collection of procedures of a low level of explicitness.  Descriptive: use different types of descriptive statistics (frequencies, central tendencies, and variabilities).  Correlational analysis: examination of the relationships between variables.  Multivariate procedures: more complex relationships, dealing with a numbers of variables at a time.
  25. 25. Experimental research procedures for analyzing data from experimental research:  T-test: helps examine whether the differences between two samples are statistically significant.  One way analysis of variance: examines differences between more than two groups;  Factorial analysis of variance: analyzing the effect of a treatment under more complex conditions.  Chi square: compare frequencies observed in a sample with some theoretically expected frequencies.
  26. 26. Using Computer for Data Analysis  The most popular is SPSS and its updated version. (the researchers are advised to find out which packages are available when preparing the research proposal.
  27. 27. Different phases in performing computer data analysis  Phase 1: prepare the data collection tools with a coding system integrated into the procedures.  Phase 2: the data are transferred to coding sheets. (Example see next slide)  Phase 3: the data are transferred to the computer database. (with professional helps)  Phase 4: choose an appropriate program for the analysis. (experts advise is encouraged)  Phase 5: get results.
  28. 28. Coding sheet
  29. 29. Caution for the researcher during the data analysis  Should have a “feel” for the results and to use intuition. (false results, or error)  Keep a close watch on the results (sensible).  Understand the statistics used for the data analysis.  Acquaint themselves with the specific statistical procedures.

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