MARKETING RESEARCH
LECTURE 6B.
CORRELATIONS ADD-ON.
Kendall’s tau.
The basic formula is
Tau = nc-nd / ((n(n-1)/2)
2 additional variants of tau coefficient exist – tau b and tau c. Both are similar to the original, but have built-in mechanisms to treat ties (same values).
E.g.
x1 x2 x3 x4
A 1 2 3 4
B 4 2 3 1
Respondents Xi possess 2 characteristics with different levels. Do the characteristics correlate?
Kendall’s tau checks all pairs of respondents (x1 x2, x1 x3, x1 x4, x2 x3, x2 x4, x3 x4) and checks how characteristics change. If both characteristics simultaneously increase or decrease – the pair is called concordant; if the first characteristic increases and second decreases – the pair is discordant.
The pairs x1 x2 are discordant (1 2, 4 2); x2 x3 are concordant (2 3, 2 4).
Plugging all numbers into the formula you get –2/3. Negative correlation.
If all pairs would be discordant, tau would become –1.
Spearman’s rank correlation.
D stand stands for difference in rank.
Example.
Var1 Var 2 Difference Difference squared
3 2 1 1
5 5 0 0
6 1 5 25
2 6 4 16
1 3 2 4
4 4 0 0
total 46
Rho = 1 – 6*46/6*6*6 – 6 = – 0.31
If there would be no difference in ranks (1 1, 2 2, 3 3, 4 4, 5 5, 6 6), Rho would become +1, if there would be complete rank incoherence (1 6, 2 5, 3 4, 4 3, 3 4, 2 5, 1 6), Rho would be –1. Intermediate Rho values appear b/w the two states above.
Significance coefficients.
2 methods: traditional and using permutation test.
To test whether the observed value of correlation is significantly different from zero, a researcher is to calculate the probability that it would be greater than or equal to the observed correlation, given the null hypothesis. (I.e. test on the chance to get the obtained correlation value accidentally).
Traditional approach is to compare the observed correlation with published tables for various levels of significance. This is a simple solution if the significance only needs to be known within a certain range or less than a certain value, as long as tables are available that specify the desired ranges.
E.g. for the Spearman’s rho (two-tailed).
N = the number of pairs of scores:
Precision
N 0.05 0.02 0.01
5 1 1
6 0.886 0.943 1
7 0.786 0.893 0.929
8 0.738 0.833 0.881
…
28 0.377 0.448 0.496
…
30 0.364 0.432 0.478
As seen above, with bigger number of cases smaller correlation coefficients appear to be significant.
Permutation test uses a transformation of obtained correlation into a specific distribution (e.g. transformation into t-distribution for Pearson’s correlation). Then, the transformed value is compared with table value of a well-known distribution under specified precision. With big samples correlation coefficients are transformed into scores that are normally distributed (and anything bigger than 2 is considered significant at 0.05% level).
(Look for t-distribution hand-out).
Note. You will never need to calculate correlations (and, it goes without saying, more complex statistics) by hand. Good statistical packages (SPSS, SAS, Wolfram Mathematica, etc.) produce all possible coefficients from input data and immediately indicate the level of significance (e.g., “significant at 0,000” means that the uniqueness of the coefficient is higher than 1/5000).
However, understanding the concepts is necessary for correct interpretation of results and efficient application of newly produced data.