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Statistical Analysis of Questionnaire Data for Internal Consistency Measurement

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Learn about Cronbach’s Alpha Coefficient for evaluating internal consistency of questionnaire data with examples and interpretation.

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Statistical Analysis of Questionnaire Data for Internal Consistency Measurement

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  1. IE 486 Work Analysis & Design II Instructor: Vincent Duffy, Ph.D. School of Industrial Eng. & Ag.& Bio Eng. Lab 3 – Evaluation of Questionnaire Data Friday, February 23, 2007

  2. Layout of Data Sheet People Observed Score (for each question) Total (Yi.) p1 7 6 13 p2 3 5 8 p3 3 3 6 p4 4 3 7 p5 5 5 10 p6 5 3 8 p7 6 6 12 p8 5 3 8 p9 5 5 10 p10 1 2 3 p11 7 6 13 p12 2 2 4 p13 4 5 9 Total ( Y.j ) 57 54 111 (Y..)

  3. k is the number of items in the group. s2res is the variance of residual components, which can not be controlled. s2p is the variance component for person. * The alpha coefficient is interpreted as the ratio of true score variance to observed score variance. Cronbach’s Alpha as a measure of Internal Consistency

  4. Cronbach’s Alpha as a measure of Internal Consistency Cronbach’s Alpha Coefficient Example: Number of people : N=13 Number of questions: k=2 i =1..a, j = 1..b, where a=13, b=2. SS(total) = (Yij2) – Y..2/ ab = 72+62+32+…+52 – (1112)/26 = 67.1154 SS(people) = (Yi.2) / b – Y..2/ ab = (132+82+….+92)/2 – (1112)/26 = 58.6154 SS(question) = (Y.j2) /a– Y..2/ ab = (572+542)/13 – (1112)/26 = 0.3462 SS(residual) = SS(total) – SS(people) – SS(question) = 67.1154 – 58.6154 – 0.3462 = 8.1538

  5. Cronbach’s Alpha Coefficient Example (cont’) MSp = SSp/(a-1) = 58.6154/12 = 4.8846 MSr = SSr/(ab-a-b+1) = 8.1538/12 = 0.6795 The estimated variance of person effect : S2p= [MSp – MSr ] / k = (4.8846 – 0.6795) / 2 = 2.1026 (variance component for person) The estimated variance of residual effect: s2r = MSr = 0.6795 (variance component for residual) The Alpha coefficient is calculated as: (k2 * s2p )) / (k2 * s2p + k * s2p) = 4*2.1026/(4*2.1026 + 2*0.6795) = 0.86089

  6. Exercise • Suppose we collected additional data as follows: • What is Cronbach’s alpha coefficient? • Can we conclude that our questionnaire has internal consistency.

  7. IE 486 Work Analysis & Design II Instructor: Vincent Duffy, Ph.D. School of Industrial Eng. & Ag.& Bio Eng. Lab 3 – Evaluation of Questionnaire Data Friday, February 23, 2007

  8. k is the number of items in the group. s2res is the variance of residual components, which can not be controlled. s2p is the variance component for person. * The alpha coefficient is interpreted as the ratio of true score variance to observed score variance. Cronbach’s Alpha as a measure of Internal Consistency

  9. Cronbach’s Alpha as a measure of Internal Consistency • For example, suppose 13 people were asked to rate a pair of questions on a 7-point scale. • The pair of questions look different but they are testing the same item. • For example • How much do you like the weather today? • How do you feel about the weather today?

  10. Layout of Data Sheet People Observed Score (for each question) Total (Yi.) p1 7 6 13 p2 3 5 8 p3 3 3 6 p4 4 3 7 p5 5 5 10 p6 5 3 8 p7 6 6 12 p8 5 3 8 p9 5 5 10 p10 1 2 3 p11 7 6 13 p12 2 2 4 p13 4 5 9 Total ( Y.j ) 57 54 111 (Y..)

  11. Cronbach’s Alpha as a measure of Internal Consistency Cronbach’s Alpha Coefficient Example: Number of people : N=13 Number of questions: k=2 i =1..a, j = 1..b, where a=13, b=2. SS(total) = (Yij2) – Y..2/ ab = 72+62+32+…+52 – (1112)/26 = 67.1154 SS(people) = (Yi.2) / b – Y..2/ ab = (132+82+….+92)/2 – (1112)/26 = 58.6154 SS(question) = (Y.j2) /a– Y..2/ ab = (572+542)/13 – (1112)/26 = 0.3462 SS(residual) = SS(total) – SS(people) – SS(question) = 67.1154 – 58.6154 – 0.3462 = 8.1538

  12. Cronbach’s Alpha Coefficient Example (cont’) MSp = SSp/(a-1) = 58.6154/12 = 4.8846 MSr = SSr/(ab-a-b+1) = 8.1538/12 = 0.6795 The estimated variance of person effect : S2p= [MSp – MSr ] / k = (4.8846 – 0.6795) / 2 = 2.1026 (variance component for person) The estimated variance of residual effect: s2r = MSr = 0.6795 (variance component for residual) The Alpha coefficient is calculated as: (k2 * s2p )) / (k2 * s2p + k * s2res) = 4*2.1026/(4*2.1026 + 2*0.6795) = 0.86089

  13. Cronbach’s Alpha as a measure of Internal Consistency SAS code for Cronbach’s alpha data one; input q1 q2; cards; 7 6 3 5 3 3 : : : 4 5 ; proc corr alpha; var q1 q2; run; Output from SAS: Raw value of Coefficient : 0.860892 Same as the result from hand calculation.

  14. Interpreting Cronbach’s Alpha results How to interpret the result? • The higher the correlation coefficient, the higher the internal consistency of the test. • The acceptable range for Cronbach’s alpha coefficient is usually between 0.7 – 1.0.

  15. Interpreting Cronbach’s Alpha results • What to do if the coefficient is low, such as 0.5? Check the following: • Are the questions ambiguous or not? • Are the scales sensitive enough to detect difference? Assumption: Question pairs are asked in the same direction

  16. Exercise We wanted to measure a concept: ‘the user satisfaction about a web site’. To measure that concept, we made one pair of questions. • How do you like this web site? 1) strongly dislike … 4) neutral … 7) strongly like • I’m very pleased with this web site when using it. 1) strongly disagree … 4) neutral … 7) strongly agree

  17. Exercise • Suppose we collected additional data as follows: • What is Cronbach’s alpha coefficient?

  18. Exercise • Step 1: calculate total (Yi., Y.j, Y..)

  19. Exercise • Step 2: calculate SS (a = 4, b = 2) • SS(total) = (62+72+32+22+52+72+42+52) – (392)/8 = 22.875 • SS(people) = (132+52+122+92)/2 – (392)/8 = 19.375 • SS(question) = (182+212)/4 – (392)/8 = 1.125 • SS(residual) = 22.875 – 19.375 – 1.125 = 2.375 Step 3: calculate MS • MS(people) = SS(people) / (a-1) = 19.375/(4-1) = 6.458 • MS(residual) = SS(residual)/ (ab-a-b+1) = 2.375/(8-4-2+1) = 0.792

  20. Exercise • Step 4: calculate the estimated variance of person effect and residual effect (k = b = 2) S2(people) = [MS(people) – MS(residual)] / k = (6.458 - 0.792) / 2 = 2.833 variance component for person S2(residual) = MS(residual) = 0.792 variance component for residual

  21. Exercise • Step 5: calculate the Cronbach’s alpha coefficient • Alpha coefficient = [ k2 S2(people) ] / [k2 S2(people) + kS2(residual) ] = (22 * 2.833) / (22 * 2.833 + 2 * 0.792) = 0.877 • Alpha coefficient > 0.7 • We may conclude that our questionnaire has internal c\onsistency.

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