Nested Factors in Repeated Measures Using SPSS

SPSS will not allow you to specify nested factors or random effects in a repeated measures design. To obtain the correct terms, you need to do some manipulation of the output.

For example, in class we discussed the controversy that erupted when Herb Clark pointed out that words should be treated as random effects in classic psycholinguistics experiments. A typical design for these experiments would be:

S^R(G^F) X W^R(C^F)

S: Subjects – random

G: Gender – fixed

W: Words – random

C: Classes – fixed

Source E(MS) df Error Line

1. G nwcs²_G + wcs²_S(G) + ns²_GW(C) + s²_S(G)W(C) g-1 Quasi-F
2. S(G) wcs²_S(G) + s²_S(G)W(C) (n-1)g 8
3. C nwgs²_C + ngs² _W(C) + ws²_S(G)C + s²_S(G)W(C) c-1 Quasi-F
4. W(C) ngs² _W(C) + s²_S(G)W(C) (w-1)c 8
5. GC nws²_GC + ns² _GW(C) + ws²_S(G)C + s²_S(G)W(C) (g-1)(c-1) Quasi-F
6. GW(C) ns² _GW(C) + s²_S(G)W(C) (g-1)(w-1)c 8
7. S(G)XC ws²_S(G)C + s²_S(G)W(C) (n-1)g(c-1) 8
8. S(G)XW(C) s²_S(G)W(C) (n-1)g(w-1)c

Quasi-F G

Quasi F C

Quasi F GC

But SPSS can only run S^R(G^F) X W^F XC^F

S: Subjects – random

G: Gender – fixed

W: Words – fixed

C: Classes – fixed

Source E(MS) df Error Line

1. G nwcs²_G + wcs²_S(G) g-1 2

2. S(G) wcs²_S(G) (n-1)g

3. C ngqs²_C + ws²_S(G)C c-1 9

4. W ncgs²_W + cs²_S(G)W w-1 10

5. WC ngs²_WC + s²_S(G)WC (w-1)(c-1) 11

6. GC nws²_GC + ws²_S(G)C (g-1)(c-1) 9

7. GW ncs²_GW + cs²_S(G)W (g-1)(w-1) 10

8. GCW ns²_GCW + s²_S(G)CW (g-1)(c-1)(w-1) 11

9. S(G)C qs²_S(G)C (n-1)g(c-1)

10. S(G)W cs²_S(G)W (n-1)g(w-1)

11. S(G)WC s²_S(G)CW (n-1)g(w-1)(c-1)

To rearrange the ANOVA source table output to correctly fit the intended design S(G)XW(C), recall that nesting is a type of interaction and that because of the cells that are missing from a nested design that would be present in a crossed design, nesting confounds the main effects and the interaction.

So, in the case above:

Intended Source Lines in SPSS ANOVA Table df df in ANOVA table

1. G G g-1 g-1
2. S(G) S(G) (n-1)g (n-1)g
3. C C c-1 c-1
4. W(C) W + WC (w-1)c (w-1)(c-1) + (w-1)
5. GC GC (g-1)(c-1) (g-1)(c-1)
6. GW(C) GW + GCW (g-1)(w-1)c (g-1)(w-1)+ (g-1)(c-1)(w-1)
7. S(G)XC S(G)C (n-1)g(c-1) (n-1)g(c-1)
8. S(G)XW(C) S(G)W+S(G)WC (n-1)g(w-1)c (n-1)g(w-1)+ (n-1)g(w-1)(c-1)

Notice that the df in the output will add up to what they should be in the E(MS) table for the nested design. For example for W(C) the desired df = (w-1)c. The SPSS output for W has w-1 df. The df in the SPSS output for WC = (w-1)(c-1) which = (w-1)c – (w-1). When we combine the source lines for W and WC, the df add so that the df for the combined total is (w-1) [from W] + (w-1)c-(w-1) [from WC] which equals:

(w-1) + (w-1)c – (w-1) = (w-1)c. So, checking that the df add up as you would expect from the E(MS) table is a good check on your work.

Then just proceed to calculate the F-ratios and Quasi-F ratios as specified in your E(MS) Table