Testing Mean Differences among Groups: Multivariate and Repeated Measures Analysis with Minimal Assumptions

Figures & data

Figure 1. Topographical maps of EEG activity in in frequency ranges of interest in a patient sample with AD.

Table 1. Number of observations for the different factor level combinations.

Sex	Age	AD	MCI	SCC	Σ
M	< 70	2	15	14	31
M	⩾ 70	10	12	6	28
F	< 70	9	13	29	51
F	⩾ 70	15	17	18	50
Σ		36	57	67	160

Table 2. Multivariate analysis of EEG data. Factors age (⩾ 70), diagnosis (AD, MCI, SCC), and sex. WTS is the Wald-type statistic approximated by a χ²-distribution, PBS denotes the asymptotic model-based “parametric” bootstrap.

	Test		WTS	PBS
	statistic	df	p-value	p-value
sex	15.54	6	0.0164	0.0321
age	18.69	6	0.0047	0.0093
sex*age	5.52	6	0.4792	0.5106
sex	12.60	6	0.0498	0.1132
diagnosis	55.16	12	<0.0001	0.0006
sex*diagnosis	9.79	12	0.6344	0.7462
diagnosis	41.81	12	<0.0001	0.0031
age	7.86	6	0.2488	0.3316
diagnosis*age	9.29	12	0.6777	0.7611

Table 3. Multivariate analysis of EEG data using classical methods. Factors age (⩾ 70), diagnosis (AD, MCI, SCC), and sex. PBS denotes the p-value from the parametric bootstrap of the WTS.

	Df	Pillai	Approx F	num Df	den Df	Pr(>F)	PBS p-value
sex	1	0.10784	3.0420	6	151	0.007748	0.0321
age	1	0.11469	3.2603	6	151	0.004828	0.0093
sex*age	1	0.03269	0.8505	6	151	0.533001	0.5106
sex	1	0.12615	3.5850	6	149	0.002391	0.1132
diagnosis	2	0.28963	4.2334	12	300	0.0000035	0.0006
sex*diagnosis	2	0.10375	1.3678	12	300	0.180272	0.7462
diagnosis	2	0.24986	3.5691	12	300	0.00005	0.0031
age	1	0.07388	1.9810	6	149	0.07182	0.3316
age*diagnosis	2	0.05879	0.7572	12	300	0.69442	0.7611

Table 4. Three-way layout for SPECT data. Between-subjects factors sex and age. Within-subjects factor brain region. WTS is the Wald-type statistic approximated by a χ²-distribution, PBS denotes the asymptotic model-based “parametric” bootstrap.

	Test		WTS	PBS
	statistic	df	p-value	p-value
sex	0.01	1	0.9258	0.9262
age	8.22	1	0.0042	0.0061
region	507.27	5	<0.0001	<0.0001
sex*age	0.03	1	0.8671	0.8625
sex*region	16.18	5	0.0063	0.0089
age*region	14.46	5	0.0129	0.0177
sex*age*region	8.12	5	0.1497	0.1773

Table 5. Three-way layout for SPECT data. Between-subjects factors sex and age. Within-subjects factor brain region. PBS denotes the asymptotic model-based “parametric” bootstrap, for comparison.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)	PBS p-value
sex	1	0.0126	0.0126	0.0063	0.9370	0.9262
age	1	73.3604	73.3604	36.4026	0.0000	0.0061
region	5	190.1318	38.0264	18.8693	0.0000	<0.0001
sex*age	1	0.2257	0.2257	0.1120	0.7380	0.8625
sex*region	5	14.2690	2.8538	1.4161	0.2158	0.0089
age*region	5	6.4003	1.2801	0.6352	0.6729	0.0177
sex*age*region	5	7.1688	1.4338	0.7115	0.6149	0.1773

Table 6. Multivariate analysis of SPECT data. Factors age (⩾ 70), diagnosis (AD, MCI, SCC), and sex. WTS is the Wald-type statistic approximated by a χ²-distribution, PBS denotes the asymptotic model-based “parametric” bootstrap.

	Test		WTS	PBS
	statistic	df	p-value	p-value
sex	17.24	6	0.0084	0.0127
age	21.65	6	0.0014	0.0042
sex*age	10.81	6	0.0944	0.1176
sex	14.70	6	0.0227	0.0455
diagnosis	61.55	12	<0.0001	<0.0001
sex*diagnosis	5.73	12	0.9292	0.9517
diagnosis	59.53	12	<0.0001	0.0001
age	16.36	6	0.0120	0.0258
diagnosis*age	11.01	12	0.5281	0.6419

Table 7. Three-way layout for SPECT data. Between-subjects factors sex and diagnosis. Within-subjects factor brain region. WTS is the Wald-type statistic approximated by a χ²-distribution, PBS denotes the asymptotic model-based “parametric” bootstrap.

	Test		WTS	PBS
	statistic	df	p-value	p-value
sex	0.01	1	0.9246	0.9231
diagnosis	51.23	2	<0.0001	<0.0001
region	426.56	5	<0.0001	<0.0001
sex*diagnosis	0.91	2	0.6333	0.6374
sex*region	14.16	5	0.0146	0.0264
diagnosis*region	18.31	10	0.0500	0.1119
sex*diagnosis*region	5.37	10	0.8651	0.8936

Table 8. Marginal effects/repeated measures analysis. Four-way layouts for EEG data. Between-subjects factors sex and diagnosis (AD, MCI, SCC). Within-subjects factors brain region (frontal, central, temporal) and feature (brain rate, complexity). WTS stands for the classical Wald-type statistic approximated by a χ²-distribution, whereas PBS denotes the asymptotic model-based “parametric” bootstrap procedure.

	Test		WTS	PBS
Effect	statistic	df	p-value	p-value
sex	9.97	1	0.0016	0.0048
diagnosis	42.38	2	<0.0001	<0.0001
feature	0.09	1	0.7687	0.7728
region	0.07	2	0.9658	0.9662
sex*diagnosis	3.78	2	0.1513	0.1742
sex*feature	2.17	1	0.1410	0.1524
sex*region	0.88	2	0.6454	0.6615
diagnosis*feature	5.32	2	0.0701	0.0913
diagnosis*region	6.12	4	0.1903	0.2316
feature*region	0.65	2	0.7216	0.7461
sex*diagnosis*feature	1.74	2	0.4199	0.4347
sex*diagnosis*region	1.53	4	0.8210	0.8396
sex*feature*region	0.42	2	0.8095	0.8194
diagnosis*feature*region	7.14	4	0.1286	0.1788
sex*diagnosis*feature*region	2.27	4	0.6855	0.7109