Z-test on overlapping samples
Quick reference
To request a Z-test on overlapping samples, type:
stat=z4
on the tab statement.
More information
The Z-test on proportions in overlapping samples is a table-level statistic. It is used to test differences between row percentages in a base row for a column axis with overlapping categories. For example, in a wine-tasting survey you might want to test whether the proportion of respondents trying sweet red wine is the same as the proportion trying dry white wine.
This test requires a stat=z4 option on the tab statement. The table must consist of an axis tabbed against itself, and the axis must allow multicoding. The first element of the axis must be a base element, and there must be at least two basic count elements in the axis. The test calculates a Z‑value comparing each row percentage with each of the other row percentages in the base row, and produces a triangular table showing all the Z-values and their associated significance levels. This table is labeled with the text ‘Z TEST – TYPE 4’.
Notes
▪The percentages (proportions) which are compared are always calculated by dividing the count in each element of the base row by the overall base. It is not necessary for the row percentages to be printed using the option op=0.
▪Although the test is only comparing proportions in the base row, it is necessary to have the axis tabbed against itself because Quantum needs to know the extent of the overlap between the different elements of the axis.
▪The Z-tests may give misleading results when the base from which proportions are calculated is small. In this test the base should be at least 20.
▪The calculation for Z subtracts the first proportion from the second, rather than the more usual method of subtracting the second proportion from the first.
Example
Suppose you have asked a multiple-response question about brand usage, and you want to see whether different proportions of respondents have tried the products. Because the groups of respondents who have tried different products may not be mutually exclusive, you cannot use the type-3 Z-test.
The Quantum program is:
tab brand brand;stat=z4
ttlQ7: Which of these brands have you ever tried?
ttlBase: All Respondents
l brand
col 123;Base;Washo;Suds;Gleam;Sparkle
The table produced is:
Q7: Which of these brands have you ever tried?
Base: All Respondents
Base Washo Suds Gleam Sparkle
Base 427 334 92 66 78
Washo 334 334 50 36 40
Suds 92 50 92 18 9
Gleam 66 36 18 66 12
Sparkle 78 40 9 12 78
Z TEST - TYPE 4
Washo Suds Gleam
Suds -17.610
0.000
Gleam -21.201 -2.369
0.000 0.018
Sparkle -19.160 -1.137 1.097
0.000 0.255 0.273 |
The results of this example show that:
▪The proportion of respondents who have tried Washo is highly significantly different from the proportions who have tried any of the other three brands.
▪There is a difference between the proportions who have tried Gleam and Suds, with a significance level of 1.8%. So you can be 98.2% confident that there is a difference between these two proportions.
▪You can have only low levels of confidence in the difference between the proportions for those who tried Sparkle compared to Suds and Gleam.
See also