Data from Beckham (1929). The author investigated the happiness of Black Americans.
The Cells tab is used to specify the information displayed in the contingency table. It is important that you ask for expected counts because this is how we check the assumptions about the expected frequencies. It is also useful to have a look at the row, column and total percentages because these values are usually more easily interpreted than the actual frequencies and provide some idea of the origin of any significant effects. We can also select standardized residuals to break down a significant effect (should we get one).
The chi-square test is highly significant, χ2(7) = 936.14, p < .001. This indicates that the profile of yes and no responses differed across the professions. Looking at the standardized residuals, the only profession for which these are non-significant are housewives who showed a fairly even split of whether they thought Black Americans were happy (40%) or not (60%). Within the other professions all of the standardized residuals are much higher than 1.96, so how can we make sense of the data? What’s interesting is to look at the direction of these residuals (i.e., whether they are positive or negative).
For the following professions the residual for ‘no’ was positive but for ‘yes’ was negative; these are therefore people who responded more than we would expect that Black Americans were not happy and less than expected that Black Americans were happy: college students, preachers and lawyers. In other words, people thought they were unhappier, compared to their Black American peers.
The remaining professions (labourers, physicians, school teachers and musicians) show the opposite pattern: the residual for ‘no’ was negative but for ‘yes’ was positive; these are, therefore, people who responded less than we would expect that Black Americans were not happy and more than expected that Black Americans were happy. In other words, people thought they were unhappier, compared to their Black American peers. It's worth noting that overall, the majority of responses thought that Black Americans would be unhappy.
We run this analysis in exactly the same way except that we now have to swap the variable happy to you_happy.
The chi-square test is highly significant, χ2(7) = 1390.74, p < .001. This indicates that the profile of yes and no responses differed across the professions. Looking at the standardized residuals, these are significant in most cells with a few exceptions: physicians, lawyers and school teachers saying ‘yes’. Within the other cells all of the standardized residuals are much higher than 1.96. Again, we can look at the direction of these residuals (i.e., whether they are positive or negative).
For labourers, housewives, school teachers and musicians the residual for ‘no’ was positive but for ‘yes’ was negative; these are, therefore, people who responded more than we would expect that they were not happy as Black Americans and less than expected that they were happy as Black Americans. In other words, they were generally unhappier than expected, compared to their Black American peers.
The remaining professions (college students, physicians, preachers and lawyers) show the opposite pattern: the residual for ‘no’ was negative but for ‘yes’ was positive; these are, therefore, people who responded less than we would expect that they were not happy as Black Americans and more than expected that they were happy as Black Americans. In other words, they were generally happier than expected, compared to their Black American peers.
Essentially, the former group are in low-paid jobs in which conditions would have been very hard (especially in the social context of the time). The latter group are in much more respected (and probably better-paid) professions. Therefore, the responses to this question could say more about the professions of the people asked than their views of being Black Americans.
We run this analysis in exactly the same way except that we now have to swap the variable you_happy to should_be_happy.
The chi-square test is highly significant, χ2(7) = 1784.23, p < .001. This indicates that the profile of yes and no responses differed across the professions. Looking at the standardized residuals, these are nearly all significant. Again, we can look at the direction of these residuals (i.e., whether they are positive or negative).
For college students and lawyers the residual for ‘no’ was positive but for ‘yes’ was negative; these are, therefore, people who responded more than we would expect that they thought that Black Americans should not be happy and less than expected that they thought Black Americans should be happy.
The remaining professions show the opposite pattern: the residual for ‘no’ was negative but for ‘yes’ was positive; these are, therefore, people who responded less than we would expect that they did not think that Black Americans should be happy and more than expected that they thought that Black Americans should be happy.
What is interesting here and in the first question is that college students and lawyers are in vocations in which they are expected to be critical about the world. Lawyers may well have defended Black Americans who had been the subject of injustice and discrimination or racial abuse, and college students would likely be applying their critically trained minds to the immense social injustice that prevailed at the time. Therefore, these groups can see that their racial group should not be happy and should strive for the equitable and just society to which they are entitled. People in the other professions perhaps adopt a different social comparison.
It’s also possible for this final question that the groups interpreted the question differently: perhaps the lawyers and students interpreted the question as ‘should they be happy given the political and social conditions of the time?’, while the others interpreted the question as ‘do they deserve happiness?’
It might seem strange to have picked a piece of research from so long ago to illustrate the chi-square test, but what I wanted to demonstrate is that simple research can sometimes be incredibly illuminating. This study asked three simple questions, yet the data are fascinating. It raised further hypotheses that could be tested, it unearthed very different views in different professions, and it illuminated very important social and psychological issues. There are others studies that sometimes use the most elegant paradigms and the highly complex methodologies, but the questions they address are meaningless for the real world. They miss the big picture. Albert Beckham was a remarkable man, trying to understand important and big real-world issues that mattered to hundreds of thousands of people.