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Ton J. Cleophas and Aeilko H. ZwindermanSPSS for Starters and 2nd Levelers10.1007/978-3-319-20600-4_11

11. Doubly Repeated Measures Analysis of Variance (16 Patients)

Ton J. Cleophas^{1, 2} and Aeilko H. Zwinderman^2,
3

(1)

Department Medicine, Albert Schweitzer Hospital, Dordrecht, The Netherlands

(2)

European College Pharmaceutical Medicine, Lyon, France

(3)

Department Biostatistics, Academic Medical Center, Amsterdam, The Netherlands

This chapter was previously partly published in “Machine learning in medicine a complete overview” as Chap. 45, 2015.

1 General Purpose

Repeated-measures ANOVA, as reviewed in the Chaps. 9 and 10, uses repeated measures of a single outcome variable in a single subject. If a second outcome variable is included and measured in the same way, the doubly-repeated-measures analysis of variance (ANOVA) procedure, available in the general linear models module, will be adequate for analysis.

2 Schematic Overview of Type of Data File

3 Primary Scientific Question

Can doubly-repeated-measures ANOVA be used to simultaneously assess the effects of three different treatment modalities on two outcome variables, and include predictor variables in the analysis.

4 Data Example

Morning body temperatures in patients with sleep deprivation is lower than in those without sleep deprivation. In 16 patients a three period crossover study of three sleeping pills (treatment levels) were studied. The underneath table give the data of the first 8 patients. The entire data file is entitled “chapter11doublyrepeatedmeasuresanova”, and is in extras.springer.com. Two outcome variables are measured at three levels each. This study would qualify for a doubly multivariate analysis.

5 Doubly Repeated Measures ANOVA

We will start by opening the data file in SPSS. For analysis the statistical model Repeated Measures in the module General Linear Model is required.

Command:

Analyze....General Linear Model....Repeated Measures....Within-Subject Factor Name: type treatment....Number of Levels: type 3....click Add....Measure Name: type hours....click Add....Measure Name: type temp....click Add....click Define ....Within-Subjects Variables(treatment): enter hours a, b, c, and temp a, b, c.... Between-Subjects Factor(s): enter gender....click Contrast....Change Contrast ....Contrast....select Repeated....click Change....click Continue....click Plots.... Horizontal Axis: enter treatment....Separate Lines: enter gender....click Add....click Continue....click Options....Display Means for: enter gender*treatment....mark Estimates of effect size....mark SSCP matrices....click Continue....click OK.

The underneath table is in the output sheets.

Multivariate tests^a

Effect			Value	F	Hypothesis df	Error df	Sig.	Partial Eta squared
Between subjects	Intercept	Pillai’s trace	1,000	3,271E6	2,000	13,000	,000	1,000
		Wilks’ lambda	,000	3,271E6	2,000	13,000	,000	1,000
		Hotelling’s trace	503211,785	3,271E6	2,000	13,000	,000	1,000
		Roys largest root	503211,785	3,271E6	2,000	13,000	,000	1,000
	Gender	Pillai’s trace	,197	1,595^b	2,000	13,000	,240	,197
		Wilks’ lambda	,803	1,595^b	2,000	13,000	,240	,197
		Hotelling’s trace	,245	1,595^b	2,000	13,000	240	,197
		Roys largest root	,245	1,595^b	2,000	13,000	240	,197
Within subjects	Treatment	Pillai’s trace	,562	3,525^b	4,000	11,000	,044	,562
		Wilks’ lambda	,438	3,525^b	4,000	11,000	,044	,562
		Hotelling’s trace	1,282	3,525^b	4,000	11,000	,044	,562
		Roys largest root	1,282	3,525^b	4,000	11,000	,044	,562
	Treatment * gender	Pillai’s trace	,762	8,822^b	4,000	11,000	,002	,762
		Wilks’ lambda	,238	8,822^b	4,000	11,000	,002	,762
		Hotelling’s trace	3,208	8,822^b	4,000	11,000	,002	,762
		Roys largest root	3,208	8,822^b	4,000	11,000	,002	,762

^aDesign: Intercept + gender. Within subjects design: treatment

^bExact statistic

Doubly multivariate analysis has two sets of repeated measures plus separate predictor variables. For analysis of such data both between and within subjects tests are performed. We are mostly interested in the within subject effects of the treatment levels, but the above table starts by showing the not so interesting gender effect on hours of sleep and morning temperatures. They are not significantly different between the genders. More important is the treatment effects. The hours of sleep and the morning temperature are significantly different between the different treatment levels at p = 0,044. Also these significant effects are different between males and females at p = 0,002.

Tests of within-subjects contrasts

Source	Measure	treatment	Type III sum of squares	df	Mean square	F	Sig.	Partial Eta squared
Treatment	Hours	Level 1 vs. Level 2	,523	1	,523	6,215	,026	,307
	Hours	Level 2 vs. Level 3	62,833	1	62,833	16,712	,001	,544
	Temp	Level 1 vs. Level 2	49,323	1	49,323	15,788	,001	,530
	Temp	Level 2 vs. Level 3	62,424	1	62,424	16,912	,001	,547
Treatment * gender	Hours	Level 1 vs. Level 2	,963	1	,963	11,447	,004	,450
		Level 2 vs. Level 3	,113	1	,113	,030	,865	,002
	Temp	Level 1 vs. Level 2	,963	1	,963	,308	,588	,022
	Temp	Level 2 vs. Level 3	,054	1	,054	,015	,905	,001
Error(treatment)	Hours	Level 1 vs. Level 2	1,177	14	,084
	Hours	Level 2 vs. Level 3	52,637	14	3,760
	Temp	Level 1 vs. Level 2	43,737	14	3,124
	Temp	Level 2 vs. Level 3	51,676	14	3,691

The above table shows, whether differences between levels of treatment were significantly different from one another by comparison with the subsequent levels (contrast tests). The effects of treatment levels 1 versus (vs) 2 on hours of sleep were different at p = 0,026, levels 2 vs 3 at p = 0,001. The effects of treatments levels 1 vs 2 on morning temperatures were different at p = 0,001, levels 2 vs 3 on morning temperatures were also different at p = 0,001. The effects on hours of sleep of treatment levels 1 vs 2 accounted for the differences in gender remained very significant at p = 0,004.

Gender * treatment

Measure	Gender	Treatment	Mean	Std. Error	95 % confidence Interval
Measure	Gender	Treatment	Mean	Std. Error	Lower bound	Upper bound
hours	,00	1	6,980	,268	6,404	7,556
		2	7,420	,274	6,833	8,007
		3	5,460	,417	4,565	6,355
	1,00	1	7,350	,347	6,607	8,093
		2	7,283	,354	6,525	8,042
		3	5,150	,539	3,994	6,306
temp	,00	1	37,020	,284	36,411	37,629
		2	35,460	,407	34,586	36,334
		3	37,440	,277	36,845	38,035
	1,00	1	37,250	,367	36,464	38,036
		2	35,183	,526	34,055	36,311
		3	37,283	,358	36,515	38,051

The above table shows the mean hours of sleep and mean morning temperatures for the different subsets of observations. Particularly, we observe the few hours of sleep on treatment level 3, and the highest morning temperatures at the same level. The treatment level 2, in contrast, pretty many hours of sleep and, at the same time, the lowest morning temperatures (consistent with longer periods of sleep). The underneath figures show the same.

6 Conclusion

Doubly multivariate ANOVA is for studies with multiple paired observations with more than a single outcome variable. For example, in a study with two or more different outcome variables the outcome values are measured repeatedly during a period of follow up or in a study with two or more outcome variables the outcome values are measured at different levels, e.g., different treatment dosages or different compounds. The multivariate approach prevents the type I errors from being inflated, because we only have one test and, so, the p-values need not be adjusted for multiple testing (see references in the underneath section).

7 Note

More background, theoretical and mathematical information of multiple treatments and multiple testing is given in “Machine learning in medicine part three, the Chap. 3, Multiple treatments, pp 19–27, and the Chap. 4, Multiple endpoints, pp 29–36, 2013, Springer Heidelberg Germany”, from the same authors.

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