DTC Quantitative Methods

Three (or more) Variables: Extensions to Analyses UsingCross- tabulations or ANOVA

Thursday 1st March 2012  

Multivariate analysis

• So far we have tended to concentrate on two-way relationships (e.g. between gender and participation in sports). But we have started to look at about three-way relationships (e.g. the gendering of the relationship between age and participation in sports).

• Social relationships and phenomena are usually more complex than is allowed for in a bivariate analysis.

• Multivariate analyses are thus commonly used as a reflection of this complexity.

• Hence, this week we will look briefly about the rationale for multivariate analysis and have a think about both cross-tabular and Analysis of Variance (ANOVA) techniques for conducting this form of analysis.

Multivariate analysis

De Vaus (1996: 198) suggests that we can use multivariate analysis to elaborate bivariate relationships, in order to answer the following questions:

1. Why does the relationship [between two variables] exist? What are the mechanisms and processes by which one variable is linked to another?

2. What is the nature of the relationship? Is it causal or non-causal?3. How general is the relationship? Does it hold for people in general,

or is it specific to certain subgroups?

This is because multivariate analysis enables the identification of:Spurious relationshipsIntervening variables

The replication of relationshipsThe specification of relationships

Age

Height Reading abilitySpurious relationship

Spurious relationships

• A spurious relationship exists where two variables are not related but a relationship between them is generated by their relationships with a third variable.

• For example:

Intervening variables• Sometimes, although there is a real (non-spurious) relationship

between two variables, we want to establish why that relationship exists.

• For example, if we discover that there is a relationship between risk of unemployment and ethnicity, we want to know why that is the case. One possibility is that some ethnic groups have lower educational levels and that this has implications for their ability to get work. In this case education would be an intervening variable.

• Intervening variables enable us to answer questions about the bivariate relationship between two variables – suggesting that (in this case) the relationship between ethnicity and unemployment is not direct but (at least in part) occurs via educational levels.

EducationEthnicity Unemployment

Is it spurious or intervening?When we do statistical tests we will obtain similar results for a spurious variable and an intervening variable: In both cases the effect of the independent variable on the dependent variable will be moderated by the third variable. So how do we know whether this third variable provides evidence of a spurious relationship or is an intervening variable?– There is no hard-and-fast statistical rule for deciding this.– But if we are suggesting that a variable is intervening, the logic of

the process must make sense – i.e. you must have a cogent theoretical reason for thinking that your independent variable affects the intervening variable which in turn affects the dependent variable.

– This kind of causal process is easiest to argue for when the timing of events supports it, i.e. when the intervening variable can be seen to occur in between the independent and dependent variables (e.g. education in the earlier example of the relationship between ethnicity and unemployment).

Replication• Sometimes when we have found a basic

(‘zero-order’) relationship between two variables (e.g. ethnicity and unemployment), we want to demonstrate that this relationship exists within different subgroups of the population (e.g. for both men and women; for those of different ages…).

• Where the relationship is replicated we can rule out the possibility that it is produced by the variable in question, either as an intervening variable or in a spurious way.

Specification

• Sometimes a particular variable only has an effect in specific situations. The variable that determines these situations is said to interact with the independent variable.

• For example, an example in De Vaus’s book suggests that going to a religious school makes boys more religious but has little or no effect on girls.

• In this case type of school interacts with gender: religious education only affects students’ religiosity in combination with being male.

Specification (interactions)

Not at all Very

How religious was your education?

Religiousness

high

low

boys

girls

Not at all Very

How religious was your education?

Religiousness

high

low

boys

girls

Interaction between No interaction sex and religiousness

of school

Graphical representation of the relationship between religious educationand religiousness, controlling for sex:

Using Cramér’s V to classify a multivariate situation

• If the Cramér’s V values for the layers are all similar, then we have a situation of replication.

• If the Cramér’s V values are smaller for the layered cross-tabulation than the value for the original cross-tabulation, then we either have a situation where the third variable is acting as an intervening variable, or one where it is inducing a spurious relationship between the original two variables. Deciding between these two options involves reflecting on whether the third variable makes sense conceptually as part of some causal mechanism linking the original two variables.

If we use SPSS to produce a cross-tabulation of two variables, then we can elaborate this relationship by introducing a third variable as a layer variable. Examining the Cramér’s V values for the original cross-tabulation and for the layers of the elaborated cross-tabulation tells us what kind of situation we are looking at:

• If the Cramér’s V values for the layered cross-tabulation vary in size, perhaps with some being smaller than the original value and some being as large or larger than it, then the situation is one of specification.

• However, if one or more of the Cramér’s V values is larger than the original value, then a failure to take account of the third variable in the first instance may also have been suppressing an underlying relationship between the two variables.

• This latter situation is a variation on the theme of spuriousness: in this case, the absence of a bivariate relationship is spurious rather than the presence of one!)

Using Cramér’s V to classify a multivariate situation (continued)

• Multivariate analyses can utilise a variety of techniques (depending on the form of the data, research questions to be addressed, etc. – we will be looking at multiple (linear) regression, but other ‘popular’ techniques include logistic regression and log-linear models), in order to determine whether the relationship between two variables persists or is altered when we ‘control for’ a third (or fourth, or fifth...) variable.

• Multivariate analysis can also enable us to establish which variable(s) has/have the greatest impact on a dependent variable – e.g. Is sex more important than ‘race’ in determining income?

• It is often important for a multivariate analysis to check for interactions between the effects of independent variables, as discussed earlier under the heading of specification.

More generally…

An example (from BSA 2006)

View on whether pre-marital sex wrong

Always Mostly Sometimes Rarely Not at all Total

Has religion? No 7 15 42 50 362 4761.5% 3.2% 8.8% 10.5% 76.1%

100.0%Yes 58 60 90 57 291 556

10.4% 10.8% 16.2% 10.3% 52.3%100.0%

Total 65 75 132 107 6531032 6.3% 7.3% 12.8% 10.4% 63.3%100.0%

24 = 86.97 (p < 0.001) Cramér’s V = 0.290

But if we split the crosstabulation by age...

Under 45: 24 = 46.66 (p < 0.001)

Cramér’s V = 0.320

45 or over: 24 = 26.00 (p < 0.001)

Cramér’s V = 0.213

Hence there is an extent to which part of the bivariate relationship was a spurious consequence of age (since 46.66 + 26.00 = 72.66, which is less than 86.97, and the Cramér’s V values show elements both of replication (since there is a statistically significant relationship for both age groups), and also of specification (since the relationship appears weaker for the younger age group, i.e. the effects of religion and age interact).

Testing for interactions

• Unfortunately, as mentioned last week, testing for an interaction in a three-way cross-tabulations requires knowledge of an additional technique (log-linear models).

• Testing for an interaction within an Analysis of Variance involving one dependent variable and two independent variables (Two-way ANOVA) is rather more straightforward…

Starting with some means…

BSA 2006: At what age did you retire work? (Q296)

NS- SEC class    N MeanEmployers in large org.; higher manag. & pr. 64 60.84Lower profess & manag; higher techn. & su. 183 58.01Intermediate occupations 88 56.18Employers in small org.; own account work 72 61.39Lower supervisory & technical occupation 96 60.04Semi-routine occupations 144 58.53Routine occupations 111 57.60

Total 758 58.65

… and then a One-Way ANOVA

BSA 2006: At what age did you retire work? (Q296)Sum of Squares   df Mean Square     F  Sig.

Between Groups 1769.833 6 294.972 3.845 .001Within Groups 57609.915 751 76.711Total 59379.748 757

Since p=0.001 < 0.05, there is a significant relationship between occupational class (NS-SEC) and retirement age.

… but we need to remember to reflect on whether the assumptions of ANOVA are met in this case!

Assumptions: a reminder• ANOVA make an assumption of homogeneity of variance (i.e. that the spread of values is the same in each of the groups).

• Furthermore, ANOVA assumes that the variable has (approximately) a normal distribution within each of the groups.

• Levene’s test of the former assumption results in p<0.001, i.e. the assumption is not plausible.

• … and it is also not self-evident that retirement ages would have a normal distribution!

Nevertheless…

• We might ask ourselves the question whether some of the class difference in retirement ages reflects gender.

• And hence there is a motivation to carry out a Two-way ANOVA to look at the effects of class and gender simultaneously.

Two-way ANOVA results

BSA2006: At what age did you retire work Q296(Type III)

Source  Sum of Sq.    df Mean Sq.       F   Sig.Corrected Model 4739.996 13 364.615 4.965 .000RClass 619.086 6 103.181 1.405 .210

RSex 2188.093 1 2188.093 29.794 .000

RClass * RSex 506.510 6 84.418 1.149 .332

Error 54639.752 744 73.441

Corrected Total 59379.748 757

… so what do the results mean?• The overall variation explained by the two variables is

greater (4739.996 compared to 1769.833).• But the between-groups variation which is unique to

class is no longer significant (p=0.210 > 0.05)• Whereas the between-groups variation which is unique

to sex is significant (p<0.001)• … but sex and class do not have interacting effects

(p=0.332)• Note that the class, sex and interaction sums of squares

don’t add up to the overall ‘explained’ sum of squares because some of the effects of class and sex overlap.

A multivariate conclusion!

• The class differences in retirement age observed in the One-way ANOVA are shown by the Two-way ANOVA to be a spurious consequence of the relationships between gender and class and between gender and retirement age!