Association between Association between Variables Measured at Variables Measured at

the Nominal Levelthe Nominal Level

IntroductionIntroduction

►The measures of association are more The measures of association are more efficient methods of expressing an efficient methods of expressing an association than calculating percentages association than calculating percentages for bivariate tables—they express the for bivariate tables—they express the relationship in a single numberrelationship in a single number

►However, you always need to look at the However, you always need to look at the bivariate tables (crosstabs), since a bivariate tables (crosstabs), since a single number loses some informationsingle number loses some information

Many Different Measures of Many Different Measures of AssociationAssociation

► Different ones are used for different levels of Different ones are used for different levels of measurement (nominal, ordinal, or measurement (nominal, ordinal, or interval/ratio)interval/ratio)

► When selecting measures of association for When selecting measures of association for assessing the relationship between variables assessing the relationship between variables measured at different levels, social scientists measured at different levels, social scientists generally choose the measure that is generally choose the measure that is appropriate for the lower of the levelsappropriate for the lower of the levels So if one variable is nominal, and the other So if one variable is nominal, and the other

interval, you would use a level of association interval, you would use a level of association appropriate for the nominal variableappropriate for the nominal variable

Chi-Square-Based Measures of Chi-Square-Based Measures of AssociationAssociation

►These have been commonly used, since These have been commonly used, since you already have calculated chi square for you already have calculated chi square for inferential statistics; it is simple to inferential statistics; it is simple to transform it into a measure of associationtransform it into a measure of association

►We can see from the percentages in a We can see from the percentages in a bivariate table that two variables are bivariate table that two variables are associated, and know from chi square that associated, and know from chi square that the differences are statistically significantthe differences are statistically significant

PhiPhi

►To find the strength of the association, To find the strength of the association, will compute a will compute a phiphi

►This statistic is used as a measure of This statistic is used as a measure of association appropriate for tables with association appropriate for tables with only two rows and two columnsonly two rows and two columns

►Formula 14.1 for phi:Formula 14.1 for phi:

2

N

Phi, cont.Phi, cont.

►Phi is the square root of the value of the Phi is the square root of the value of the obtained chi square divided by the sample obtained chi square divided by the sample sizesize

►For a 2 x 2 table, phi ranges in value from For a 2 x 2 table, phi ranges in value from 0 (no association) to 1.00 (perfect 0 (no association) to 1.00 (perfect association)association)

►A phi of .33 indicates a weak to moderate A phi of .33 indicates a weak to moderate relationship between the two variablesrelationship between the two variables

►This measure does not reveal the pattern This measure does not reveal the pattern of the association, so need to look at the of the association, so need to look at the tabletable

CramerCramer’’s Vs V

►For tables with three or more columns For tables with three or more columns or three or more rows, phi has an or three or more rows, phi has an upper limit that can exceed 1.00upper limit that can exceed 1.00 CramerCramer’’s V is used for tables that are s V is used for tables that are

larger than 2 x 2, is based on chi square, larger than 2 x 2, is based on chi square, and is also easy to calculateand is also easy to calculate

Formula for CramerFormula for Cramer’’s Vs V

1,1.min

2

crofNV

Interpretation of CramerInterpretation of Cramer’’s Vs V

► It has an upper limit of 1.00 for any size It has an upper limit of 1.00 for any size tabletable

►Like phi, it can be interpreted as an Like phi, it can be interpreted as an index that measures the strength of the index that measures the strength of the association between two variablesassociation between two variables

►A major problem with phi and CramerA major problem with phi and Cramer ’’s s V is the absence of a direct or V is the absence of a direct or meaningful interpretation for values meaningful interpretation for values between the extremes of 0.00 and 1.00between the extremes of 0.00 and 1.00 Both indicate the strength of the Both indicate the strength of the

associationassociation But it is only an index of relative strength But it is only an index of relative strength

Proportional Reduction in Error Proportional Reduction in Error (PRE)(PRE)

►For nominal-level variables, the logic For nominal-level variables, the logic of PRE involves first attempting to of PRE involves first attempting to guess or predict the category into guess or predict the category into which each case will fall on the which each case will fall on the dependent variable (Y) while ignoring dependent variable (Y) while ignoring the independent variable (X)the independent variable (X) Will be predicting blindly in this case, and Will be predicting blindly in this case, and

will make many errorswill make many errors►The second step would be to predict The second step would be to predict

again the category of each case on the again the category of each case on the dependent variable, but take the dependent variable, but take the independent variable into accountindependent variable into account

PRE, cont.PRE, cont.

► If the two variables are associated, the If the two variables are associated, the additional information from the independent additional information from the independent variable should enable us to reduce our variable should enable us to reduce our errors of predictionerrors of prediction

► The stronger the association between the The stronger the association between the variables, the more we will reduce our errorsvariables, the more we will reduce our errors In the case of a perfect association, we would In the case of a perfect association, we would

make no errors at all when predicting scores on Y make no errors at all when predicting scores on Y from scores on X from scores on X

When there is When there is nono association between the association between the variables, knowledge of the independent will not variables, knowledge of the independent will not improve the accuracy of our predictions—we improve the accuracy of our predictions—we would make just as many errors of predictionwould make just as many errors of prediction

Lambda Lambda

► Lambda is a PRE measure for nominal-level Lambda is a PRE measure for nominal-level variablesvariables

► We know that gender and height are associated We know that gender and height are associated by looking at the percentagesby looking at the percentages

► To measure the strength of this association, a To measure the strength of this association, a PRE measure called lambda will be calculatedPRE measure called lambda will be calculated First need to find the number of prediction errors First need to find the number of prediction errors

made while ignoring the independent variable made while ignoring the independent variable (gender)(gender)

Then will find the number of prediction errors made Then will find the number of prediction errors made while taking gender into accountwhile taking gender into account

These two sums will be compared in order to derive These two sums will be compared in order to derive the statisticthe statistic

Example of Height by Gender Example of Height by Gender (Table 12.15)(Table 12.15)

► Can ignore information given by the Can ignore information given by the independent variable (gender) by working independent variable (gender) by working only with the row marginalsonly with the row marginals Two different predictions can be made using Two different predictions can be made using

these marginalsthese marginals►We can predict either that all subjects are tall or that all We can predict either that all subjects are tall or that all

subjects are short (these are the only two permitted by subjects are short (these are the only two permitted by lambda)lambda)

For the first prediction (all subjects are tall), 48 For the first prediction (all subjects are tall), 48 errors will be madeerrors will be made►For this prediction, all 100 cases would be placed in the For this prediction, all 100 cases would be placed in the

first rowfirst row►Since only 52 cases belong in this row, this prediction Since only 52 cases belong in this row, this prediction

would result in (100 – 52) or 48 errorswould result in (100 – 52) or 48 errors

Example, cont.Example, cont.

► If we had predicted that all subjects If we had predicted that all subjects were short, we would have made 52 were short, we would have made 52 errors (100 -48 = 52)errors (100 -48 = 52)

►We will use the lesser of these two We will use the lesser of these two numbers and refer to this quantity as E numbers and refer to this quantity as E sub 1 for the number of errors made sub 1 for the number of errors made while ignoring the independent while ignoring the independent variablevariable So, E sub 1 = 48 [N – (largest row total)]So, E sub 1 = 48 [N – (largest row total)]

Second StepSecond Step

► The second step in computing lambda is to The second step in computing lambda is to again predict scores on Y (height), this time again predict scores on Y (height), this time taking X (gender) into accounttaking X (gender) into account Follow the same procedure as in the first step, Follow the same procedure as in the first step,

but this time move from column to columnbut this time move from column to column►Since each column is a category of X, we take X into Since each column is a category of X, we take X into

account in making our predictionsaccount in making our predictions For the left-hand column (males), we predict that For the left-hand column (males), we predict that

all 50 cases will be tall and make six errorsall 50 cases will be tall and make six errors For the second column (females), our prediction For the second column (females), our prediction

is that all females are short, and eight errors will is that all females are short, and eight errors will be madebe made

We have made a total of 14 errors of prediction, We have made a total of 14 errors of prediction, a quantity we will label E sub 2a quantity we will label E sub 2

Logic of LambdaLogic of Lambda

►The logic of lambda is that, if the The logic of lambda is that, if the variables are associated, fewer errors variables are associated, fewer errors will be made under the second will be made under the second procedure than under the first (want E procedure than under the first (want E sub 2 to be less than E sub 1)sub 2 to be less than E sub 1) Clearly, gender and height are associated, Clearly, gender and height are associated,

since we made fewer errors of prediction since we made fewer errors of prediction while considering gender (E sub 2 = 14) while considering gender (E sub 2 = 14) than while ignoring gender (E sub 1 = 48)than while ignoring gender (E sub 1 = 48)

Computing LambdaComputing Lambda

►To find the proportional reduction in To find the proportional reduction in error, use Formula 12.3:error, use Formula 12.3:

E E

E1 2

1

Interpretation of LambdaInterpretation of Lambda

►For the above example, lambda For the above example, lambda equals .71equals .71

►Lambda has a possible range of 0 to 1Lambda has a possible range of 0 to 1 A lambda of 0 would indicate that the A lambda of 0 would indicate that the

information given by the independent information given by the independent variable does not improve our ability to variable does not improve our ability to predict the dependent and therefore, that predict the dependent and therefore, that there is no association between the variablesthere is no association between the variables

A lambda of 1.00 would mean that it was A lambda of 1.00 would mean that it was possible to predict Y without error from Xpossible to predict Y without error from X

PRE InterpretationPRE Interpretation

► Additionally, lambda allows a direct and Additionally, lambda allows a direct and meaningful interpretation of the numbers in meaningful interpretation of the numbers in betweenbetween When multiplied by 100, the value of lambda When multiplied by 100, the value of lambda

indicates directly the proportional reduction in indicates directly the proportional reduction in error—the strength of the associationerror—the strength of the association

So, a lambda of .71 tells us that knowledge of So, a lambda of .71 tells us that knowledge of gender improves our ability to predict height by a gender improves our ability to predict height by a factor of 71%factor of 71%►Of, we are 71% better off knowing gender when Of, we are 71% better off knowing gender when

attempting to predict height than we are not knowing attempting to predict height than we are not knowing gendergender

Other ExamplesOther Examples

► If lambda = .20, this indicates that we are 20% If lambda = .20, this indicates that we are 20% better off knowing the independent variable better off knowing the independent variable when attempting to predict a personwhen attempting to predict a person’’s score or s score or value on the dependent variablevalue on the dependent variable

► If we make 75 errors when predicting Y without If we make 75 errors when predicting Y without knowledge of X, and 60 errors when predicting knowledge of X, and 60 errors when predicting Y with knowledge of X, then X and Y are Y with knowledge of X, then X and Y are associatedassociated

► If the value of lambda is relatively low, we may If the value of lambda is relatively low, we may conclude that other variables are importantly conclude that other variables are importantly associated with the dependent variableassociated with the dependent variable

Problems with LambdaProblems with Lambda

► It changes if you reverse the It changes if you reverse the independent and dependent variablesindependent and dependent variables Need to follow the convention of putting Need to follow the convention of putting

the independent variable in the columns the independent variable in the columns and compute lambda as done aboveand compute lambda as done above

You also need to be confident which You also need to be confident which variable is the independent one and which variable is the independent one and which is the dependent oneis the dependent one

Second ProblemSecond Problem

► If one of the row totals is much larger than the If one of the row totals is much larger than the others, lambda can take on a value of 0 even others, lambda can take on a value of 0 even when other measures of association would not when other measures of association would not be 0, and calculating percentages for the table be 0, and calculating percentages for the table indicates some association between the indicates some association between the variablesvariables Suggests that you use great caution in interpretation Suggests that you use great caution in interpretation

of lambda when the row marginals are very unequalof lambda when the row marginals are very unequal If the row totals are unequal, you should use a chi-If the row totals are unequal, you should use a chi-

square-based measure of association (phi or Cramersquare-based measure of association (phi or Cramer ’’s s V)V)

For the same bivariate table, CramerFor the same bivariate table, Cramer ’’s V is .27 and s V is .27 and lambda is zero, we can conclude that the variables lambda is zero, we can conclude that the variables may be associated even if lambda is zero—need to may be associated even if lambda is zero—need to disregard lambda if the row marginals are very disregard lambda if the row marginals are very unequalunequal