Statistics for Librarians, Session 3: Inferential statistics
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Third in a series of four seminars presented to University of North Texas librarians. This presentation focuses on using basic tests that determine the association of two sets of data based on measures of central tendency and variation.
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Statistics for Librarians, Session 3: Inferential statistics
- 3. SESSION OBJECTIVES Purpose of Inferential Statistics Probability Elements of Significance Testing Three key tests • T-test • Chi-squared • Correlation (or binomial) Effect Measures
- 4. PURPOSE OF INFERENTIAL STATISTICS • Infer results • Draw conclusions • Increase the Signal-Noise ratio Signal Noise
- 5. INFERENTIAL STATISTICS Tests of hypotheses • Expectations • Associations Accounts for uncertainty • Random error • Confidence interval
- 7. NOT TO PROVE, BUT TO FALSIFY H1 Difference H0 No Difference
- 8. NOT TO PROVE, BUT TO FALSIFY H1 >=10% Increase H0 <10% Increase
- 10. LEVELS OF MEASUREMENT (NOIR) Nominal •Counts by category •Binary (Yes/No) •No meaning between the categories (Blue is not better than Red) Ordinal •Ranks •Scales •Space between ranks is subjective Interval •Integers •Zero is just another value – doesn’t mean “absence of” •Space between values is equal and objective, but discrete Ratio •Interval data with a baseline •Zero (0) means “absence of” •Space between is continuous •Includes simple counts
- 12. CENTRAL TENDENCY BY LEVELS OF MEASUREMENT Interval or Ratio Mean Median Nominal or Rank Mode Median (rank only)
- 13. SPREAD Interval & Ratio •Range •Quantiles •Standard Deviation Nominal & Rank •Distribution Tables •Bar Graphs How variable is the data?
- 16. PROBABILITY WHAT’S PROBABILITY GOT TO DO WITH STATISTICS?
- 17. WHAT IS PROBABILITY? Chance of something happening (x) Expressed as P(x)=y Between 0 and 1 Based on distribution of events
- 18. STEM-AND-LEAF Stem Leaf 0 0111222222222222223333334444555666 6677788899 1 0000000011122223333356778899 2 00122234444799 3 0245 Groups Last digit Years at UNT 0 5 13 1 6 13 1 6 13 1 6 13 2 6 15 2 6 16 2 7 17 2 7 17 2 7 18 2 8 18 2 8 19 3 11 29 4 11 29 4 12 30 4 12 32 4 12 34 5 12 35 5 13
- 19. Stem Leaf Count 0 1122223334445555666666677777899 31 1 000011122222222333346677889 27 2 0122234468 10 3 1112355888 11 4 12 2 Range Count 0-9 31 10-19 27 20-29 10 30-39 11 40-49 2 0 10 20 30 40 0-9 10-19 20-29 30-39 40-49 Histogram of Years at UNT
- 22. Set the mean to 0 Standard Deviations above and below the mean
- 23. DEMONSTRATION OF DISTRIBUTIONS Distribution of the Population The “Truth” N is the # of samples n is the number of items in each sample Watch the cumulative mean & medians slowly merge to the population
- 24. ACTIVITIES
- 25. CASE STUDY • Background: Info-Lit course is meeting resistance from skeptical faculty. • Research Questions: • Does the IL course improve grades on final papers? • Can the IL course improve passing rates for the course? • Do students in different majors respond differently to the IL training? • Is the final score related to the number of credit hours enrolled for each student?
- 26. METHODOLOGY Selection • Two sections of same course with different instructors. • Random Assignment Outcome • Blinded scoring by 2 TAs • Scores range from 1-100 • Passing grade: 70
- 27. ACTIVITIES Table 1 • Distribution of scores Table 2 • Distribution of passing rates by major Table 3 • Correlation of scores with credit hours
- 28. DESCRIPTIVE STATISTICS OF CASE STUDY
- 29. DISTRIBUTION OF SCORES Table 1 •Distribution of scores Table 2 •Distribution of passing rates by broad field of major Table 3 •Correlation of scores & credit hours
- 31. SIGNIFICANCE TESTING • Groups against each other • A group against the population or standard Comparing significance of differences • Risk of being wrong • Alpha (α) • Set in advance What is “significant”? • The value that the statistic must meet or exceed to be statistically significant. • Based on statistic and α Critical Value
- 32. STEPS IN SIGNIFICANCE TESTING Which Test? Calculate Statistic Critical Value of Statistic? Probability (p-value)
- 33. KEY ELEMENTS OF SIGNIFICANCE TESTING Null Hypothesis Measure of Central Tendency Standard deviations Risk of being wrong (alpha) • Usually .05 or .025 or .01 or .001 Degrees of freedom (df)
- 34. DEGREES OF FREEDOM Number of values in the final calculation of a statistic that are free to vary.
- 35. DEGREES OF FREEDOM EXPLAINED • All these have a mean of 5: • 5, 5, 5 • 2, 8, 5 • 3, 2, 10 • 7, 4, & ? • If 2 values are known and the mean is known, then the 3rd value is also known. • Only 2 of the 3 values are free to vary.
- 36. CALCULATING DEGREES OF FREEDOM (DF) For a single sample: • Degrees of freedom (df) for t-test = n-1 For more than one group: • df=∑(n-1) for all groups (k) • OR, ∑ n-k For comparing proportions in categories (k): • df= ∑k-1 (# of categories minus 1)
- 38. T-TEST Used with interval or ratio data Based on normal distribution Four Decisions • Paired or un-paired samples? • Equal or unequal variances (standard deviations)? • Risk? • One- or two-tail? • Direction of expected difference • Best to bet on difference in both directions (2-tail)
- 40. T-TEST FORMULA FOR UNPAIRED SAMPLES 𝑡 = 𝑥1 − 𝑥2 𝑆 𝑥1−𝑥2 Signal Noise Difference Between Group Means 𝑉𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐺𝑟𝑜𝑢𝑝𝑠
- 41. ELEMENTS OF T-TEST USING EXCEL DATA ANALYSIS TOOLPAK •UnpairedPaired or Unpaired samples? •Equal*Equal or Unequal Variances? •Data for intervention group •Data for control group Data •0Hypothesized difference •0.025 (for a 2-tail test) Alpha
- 42. T-TEST IN EXCEL
- 43. READING T-TEST RESULTS ∑(n-1) = (51-1)+(50-1) = 50+49=99 <=0.025?
- 44. IS THE DIFFERENCE SIGNIFICANT? p=0.0005
- 45. TESTING DISTRIBUTION OF NOMINAL DATA
- 46. PEARSON’S CHI-SQUARED (Χ2) GOODNESS OF FIT TEST Does an observed frequency distribution differ from an expected distribution • Observed is the sample or the intervention. • Expected is the population or the control or a theoretical distribution. • Will depend on your Null Hypothesis Nominal or categorical data • Counts by category
- 47. EXPECTED RATIOS FOR CASE STUDY Research Question: •Do students in different majors respond differently to the IL training? Null Hypothesis •The ratio of students who passed will be the same for all majors.
- 48. WHEN TO USE PEARSON’S CHI- SQUARED GOODNESS OF FIT TEST Nominal Data Sample Size • Not too large: • Sample is at most 1/10th of population • Not too small: • At least five in each of the categories for the expected group.
- 49. OBSERVED PASSING RATES BY MAJOR Major Passed Not Passed Grand Total Arts 6 7 13 Humanities 8 5 13 Social Sciences 17 10 27 STEM 20 5 25 Undeclared 16 7 23 Total 67 34 101
- 50. EXPECTED RATIOS OF PASSING RATES BY MAJOR • H0: Rates of passing will be the same for all majors. • Expected rates: 70% of class passes. • Expected ratios: 70% of each major passes. Major Passed Not Passed Grand Total Arts 11.2 (16*.7) 4.8 16 Humanities 11.2 (16*.7) 4.8 16 Social Sciences 18.2 (26*.7) 7.8 26 STEM 16.1 (23*.7) 6.9 23 Undeclared 14 (20*.7) 6 20
- 51. CHI-SQUARED GOF TEST FORMULA •χ2 = 𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑−𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 2 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 • Critical value of Chi-squared depends on degrees of freedom. • Degrees of freedom • Based on the number of categories or table cells (k) • df=k-1
- 52. CHI-SQUARED IN EXCEL What is Null Hypothesis? There is no difference between the majors regarding passing rates. What is your alpha (risk)? 0.05 Data in a summary tables? Actual Ratios Expected Ratios Excel function: =CHISQ.TEST(actual range1,expected range2) Provides a p-value 0.0000172 Is p-value <= alpha? Yes
- 53. CORRELATION OF SCORE & SEMESTER HOURS ENROLLED
- 54. STATISTICAL CORRELATION Quantitative value of relationship of 2 variables •-1 represents a perfect indirect correlation •0 represents no correlation •+1 represents a perfect direct correlation Expressed in range of -1 to +1 •How much two variables change together Based on co-variance
- 55. PEARSON’S PRODUCT MOMENT CORRELATION COEFFICIENT Most commonly used statistic Normally distributed interval or ratio data only Labeled as r Multiplication = Interaction Signal Noise 𝑟𝑥𝑦 = 𝑥 − 𝑥 𝑦 − 𝑦 𝑛 − 1 𝑠 𝑥 𝑠 𝑦
- 56. CORRELATION IN EXCEL •No correlation Null Hypothesis? •=PEARSON(range1,range2) Coefficient function (r): Does NOT have a single function to test for significance Calculate Probability: n # in sample 101 df # in sample - 2 99 alpha 0.025 for 2-Tail Test 0.025 r =PEARSON(range1,range2) 0.362287 t =r*SQRT(alpha)/SQRT(1-r^2) 3.867434 p =T.DIST.2T(t,df) 0.000197
- 57. CORRELATIONS FOR ORDINAL DATA Spearman’s ϱ (rho) •Use if there are limited ties in rank. Kendall’s τ (tau) •Use if you have a number of ties.
- 59. KNOW THE TESTS Assumptions Limitations Appropriate data type What the test tests
- 60. FACTORS ASSOCIATED WITH CHOICE OF STATISTICAL METHOD Level of Measurement What is being compared Independence of units Underlying variance in the population Distribution Sample size Number of comparison groups
- 61. USE A FLOW CHART
- 62. GOING BEYOND THE P- VALUE EFFECT SIZES
- 63. AND THE P-VALUE SAYS… Much about the distributions More about the H0 than H1 Little about size of differences
- 64. MORE USEFUL STATISTICS Effect Sizes •Tell the real story Confidence Intervals •State your certainty
- 65. EFFECT SIZES OF QUANTITATIVE DATA Differences from the mean • Standardized • weighted against the pooled (average) standard deviation • Cohen’s d Correlations • Cohen’s guidelines for Pearson’s r • r = 0.362 Effect Size r> Small .10 Medium .30 Large .50 𝑑 = 𝑥1 − 𝑥2 𝑠 𝑥1,𝑥2 𝑑 = 79.47 − 69.56 11.8036 𝑑 = 0.8392
- 66. EFFECT SIZES OF QUALITATIVE DATA Based on Contingency table • Uses probabilities • 𝑅𝑅 = 𝑎 𝑎+𝑏 𝑐 𝑐+𝑑 Relative risk • 𝑅𝑅 = 41∗65 10∗36 •RR = 1.608 •The passing rate for the intervention group was 1.6 times the passing rate for control group. RR of Case Study Pass No Pass Total Intervention a (41) b (24) a+b (65) Control c (26) d (10) c+d (36) Totals a+c (67) b+d (34) a+b+c+d (101)
- 67. CONFIDENCE INTERVALS Point estimates Intervals Based on Expressed as: •Single value •Mean •Degree of uncertainty •Range of certainty around the point estimate •Point estimate (e.g. mean) •Confidence level (usually .95) •Standard deviation •The mean score of the students who had the IL training was 79.5 with a 95% CI of 76.4 and 82.5.
- 68. CASE STUDY CONCLUSIONS • Research Questions: • Could the IL course improve grades on final papers? • Could the IL course improve passing rates for the course? • Do students in different majors respond differently to the IL training? • Is the final score related to the number of credit hours enrolled for each student? • Control for external variables