Presentation on theme: "1. Nominal Measures of Association 2. Ordinal Measure s of Association"— Presentation transcript:
11. Nominal Measures of Association 2. Ordinal Measure s of Association
2ASSOCIATION Association The strength of relationship between 2 variablesKnowing how much variables are related may enable you to predict the value of 1 variable when you know the value of anotherAs with test statistics, the proper measure of association depends on how variables are measured
3Significance vs. Association Association = strength of relationshipTest statistics = how different findings are from nullThey do capture the strength of a relationshipt = number of standard errors that separate meansChi-Square = how different our findings are from what is expected under nullIf null is no relationship, then higher Chi-square values indicate stronger relationships.HOWEVER --- test statistics are also influenced by other stuff (e.g., sample size)
4MEASURES OF ASSOCIATION FOR NOMINAL-LEVEL VARIABLES “Chi-Square Based” Measures2 indicates how different our findings are from what is expected under null2 also gets larger with higher sample size (more confidence in larger samples)To get a “pure” measure of strength, you have to remove influence of NPhiCramer's V
5PHI Phi (Φ) = 2 √ N Formula standardizes 2 value by sample size Measure ranges from 0 (no relationship) to values considerably >1(Exception: for a 2x2 bivariate table, upper limit of Φ= 1)
6PHI Example: 2 x 2 table LIMITATION OF Φ: 2=5.28 Lack of clear upper limit makes Φ an undesirable measure of association
7CRAMER’S VCramer’s V = 2√ (N)(Minimum of r-1, c-1)Unlike Φ, Cramer’s V will always have an upper limit of 1, regardless of # of cells in tableFor 2x2 table, Φ & Cramer’s V will have the same valueCramer’s V ranges from 0 (no relationship) to +1 (perfect relationship)
82-BASED MEASURES OF ASSOCIATION Sample problem 1:The chi square for a 5 x 3 bivariate table examining the relationship between area of Duluth one lives in & type of movie preference is 8.42, significant at .05 (N=100). Calculate & interpret Cramer’s V.ANSWER:(Minimum of r-1, c-1) = 3-1 = 2Cramer’s V = .21Interpretation: There is a relatively weak association between area of the city lived in and movie preference.CogdonLincoln ParkObservation HillLakesideChester ParkComedyDramaAction
92-BASED MEASURES OF ASSOCIATION Sample problem 2:The chi square for a 4 x 4 bivariate table examining the relationship between type of vehicle driven & political affiliation is 12.32, sig. at .05 (N=300). Calculate & interpret Cramer’s V.ANSWER:(Minimum of r-1, c-1) = 4 -1 = 3Cramer’s V = .12Interpretation: There is a very weak association between type of vehicle driven & political affiliation.CompactTruckSUVLuxury
10SUMMARY: 2 -BASED MEASURES OF ASSOCIATION Limitation of Φ & Cramer’s V:No direct or meaningful interpretation for values between 0-1Both measure relative strength (e.g., .80 is stronger association than .40), but have no substantive meaning; hard to interpret“Rules of Thumb” for what is a weak, moderate, or strong relationship vary across disciplinesAGAIN, THESE ARE FOR NOMINAL-LEVEL VARIABLES ONLY.
11LAMBDA (λ)PRE (Proportional Reduction in Error) is the logic that underlies the definition & computation of lambdaTells us the reduction in error we gain by using the IV to predict the DVRange 0-1 (i.e., “proportional” reduction)E1 – Attempt to predict the category into which each case will fall on DV or “Y” while ignoring IV or “X”E2 – Predict the category of each case on Y while taking X into accountThe stronger the association between the variables the greater the reduction in errors
12LAMBDA: EXAMPLE 1 Risk Classification Re-arrested Low Medium High Does risk classification in prison affect the likelihood of being rearrested after release? (2=43.7)Risk ClassificationRe-arrestedLowMediumHighTotalYes252075120No5015854090205
13LAMBDA: EXAMPLE Find E1 (# of errors made when ignoring X) E1 = N – (largest row total)= = 85Risk ClassificationRe-arrestedLowMediumHighTotalYes252075120No5015854090205
14LAMBDA: EXAMPLE Find E2 (# of errors made when accounting for X) E2 = (each column’s total – largest N in column)= (75-50) + (40-20) + (90-75) = = 60Risk ClassificationRe-arrestedLowMediumHighTotalYes252075120No5015854090205
15CALCULATING LAMBDA: EXAMPLE Calculate Lambdaλ = E1 – E2 = = 25 = 0.294EInterpretation – when multiplied by 100, λ indicates the % reduction in error achieved by using X to predict Y, rather than predicting Y “blind” (without X)0.294 x 100 = 29.4% - “Knowledge of risk classification in prison improves our ability to predict rearrest by 29%.”
16LAMBDA: EXAMPLE 2What is the strength of the relationship between citizens’ race and attitude toward police?(obtained chi square is > (2[critical])Calculate & interpret lambda to answer this questionAttitudetowards policeRaceTotalsBlackWhiteOtherPositive4015035225Negative80955523012024590455
17LAMBDA: EXAMPLE 2 E1 = N – (largest row total) 455 – 230 = 225 E2 = (each column’s total – largest N in column)(120 – 80) + (245 – 150) + (90 – 55) == 170λ = E1 – E2 = = = 0.244EINTERPRETATION:x 100 = 24.4% - “Knowledge of an individual’s race improves our ability to predict attitude towards police by 24%”Attitudetowards policeRaceTotalsBlackWhiteOtherPositive4015035225Negative80955523012024590455
18SPSS EXAMPLEIS THERE A SIGNIFICANT RELATIONSHIP B/T GENDER & VOTING BEHAVIOR?If so, what is the strength of association between these variables?ANSWER TO Q1: “YES”
19SPSS EXAMPLE ANSWER TO QUESTION 2: By either measure, the association between these variables appears to be weak
202 LIMITATIONS OF LAMBDA 1. Asymmetric Value of the statistic will vary depending on which variable is taken as independent2. Misleading when one of the row totals is much larger than the other(s)For this reason, when row totals are extremely uneven, use a chi square-based measure instead
21ORDINAL MEASURE OF ASSOCIATION GAMMAFor examining STRENGTH & DIRECTION of “collapsed” ordinal variables (<6 categories)Like Lambda, a PRE-based measureRange is -1.0 to +1.0
22GAMMA Logic: Applying PRE to PAIRS of individuals Prejudice Lower ClassMiddle ClassUpper ClassLowKennyTimKimMiddleJoeyDebRossHighRandyEricBarb
23GAMMA CONSIDER KENNY-DEB PAIR In the language of Gamma, this is a “same” pairdirection of difference on 1 variable is the same as direction on the otherIf you focused on the Kenny-Eric pair, you would come to the same conclusionPrejudiceLower ClassMiddle ClassUpper ClassLowKennyTimKimMiddleJoeyDebRossHighRandyEricBarb
24GAMMA NOW LOOK AT THE TIM-JOEY PAIR In the language of Gamma, this is a “different” pairdirection of difference on one variable is opposite of the difference on the otherPrejudiceLower ClassMiddle ClassUpper ClassLowKennyTimKimMiddleJoeyDebRossHighRandyEricBarb
25GAMMA Logic: Applying PRE to PAIRS of individuals Formula: same – differentsame + different
26GAMMA Prejudice Lower Class Middle Class Upper Class Low Kenny Tim Kim If you were to account for all the pairs in this table, you would find that there were 9 “same” & 9 “different” pairsApplying the Gamma formula, we would get:9 – 9 = = 0.0PrejudiceLower ClassMiddle ClassUpper ClassLowKennyTimKimMiddleJoeyDebRossHighRandyEricBarbMeaning: knowing how 2 people differ on social class would not improve your guesses as to how they differ on prejudice.In other words, knowing social class doesn’t provide you with any info on a person’s level of prejudice.
27GAMMA 3-case example Applying the Gamma formula, we would get: 3 – 0 = = 1.00PrejudiceLower ClassMiddle ClassUpper ClassLowKennyMiddleDebHighBarb
28Gamma: Example 1 Examining the relationship between: FEHELP (“Wife should help husband’s career first”) &FEFAM (“Better for man to work, women to tend home”)Both variables are ordinal, coded 1 (strongly agree) to 4 (strongly disagree)
29Gamma: Example 1Based on the info in this table, does there seem to be a relationship between these factors?Does there seem to be a positive or negative relationship between them?Does this appear to be a strong or weak relationship?
30GAMMADo we reject the null hypothesis of independence between these 2 variables?Yes, the Pearson chi square p value (.000) is < alpha (.05)It’s worthwhile to look at gamma.Interpretation:There is a strong positive relationship between these factors.Knowing someone’s view on a wife’s “first priority” improves our ability to predict whether they agree that women should tend home by 75.5%.
31USING GSS DATAConstruct a contingency table using two ordinal level variablesAre the two variables significantly related?How strong is the relationship?What direction is the relationship?