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Ton J. Cleophas and Aeilko H. ZwindermanSPSS for Starters and 2nd Levelers10.1007/978-3-319-20600-4_51

51. Loglinear Models, Logit Loglinear Models (445 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. 39, Springer Heidelberg Germany, 2015.

1 General Purpose

Multinomial regression is adequate for identifying the main predictors of outcome categories, like levels of injury or quality of life (QOL). An alternative approach is logit loglinear modeling. It does not use continuous predictors on a case by case basis, but rather the weighted means of subgroups formed with the help of predictors. This approach may allow for relevant additional conclusions from your data.

2 Schematic Overview of Type of Data File

3 Primary Scientific Question

Does logit loglinear modeling allow for relevant additional conclusions from your categorical data as compared to polytomous/multinomial regression?

4 Data Example

Qol	Gender	Married	Lifestyle	Age
2	1	0	0	55
2	1	1	1	32
1	1	1	0	27
3	0	1	0	77
1	1	1	0	34
1	1	0	1	35
2	1	1	1	57
2	1	1	1	57
1	0	0	0	35
2	1	1	0	42
3	0	1	0	30
1	0	1	1	34

age (years)

gender (0 = female)

married (0 = no)

lifestyle (0 = poor)

qol (quality of life levels, 1 = low, 3 = high)

The above table shows the data of the first 12 patients of a 445 patient data file of qol (quality of life) levels and patient characteristics. The characteristics are the predictor variables of the qol levels (the outcome variable). The entire data file is in extras.springer.com, and is entitled “chapter51loglinear”. We will first perform a traditional multinomial regression in order to test the linear relationship between the predictor levels and the chance (actually the odds, or to be precise logodds) of having one of the three qol levels. Start by opening SPSS, and entering the data file.

5 Multinomial Logistic Regression

For analysis the statistical model Multinomial Logistic Regression in the module Regression is required.

Command:

Analyze....Regression....Multinomial Logistic Regression....Dependent: enter “qol”.... Factor(s): enter “gender, married, lifestyle”....Covariate(s): enter “age”....click OK.

The underneath table shows the main results.

Parameter estimates

Qol^a		B	Std. error	Wald	df	Sig.	Exp(B)	95 % confidence interval for Exp (B)
Qol^a		B	Std. error	Wald	df	Sig.	Exp(B)	Lower bound	Upper bound
Low	Intercept	28,027	2,539	121,826	1	,000
	age	−,559	,047	143,158	1	,000	,572	,522	,626
	[gender = 0]	,080	,508	,025	1	,875	1,083	,400	2,930
	[gender = 1]	0^b	.	.	0	.	.	.	.
	[married = 0]	2,081	,541	14,784	1	,000	8,011	2,774	23,140
	[married = 1]	0^b	.	.	0	.	.	.	.
	[lifestyle = 0]	−,801	,513	2,432	1	,119	,449	,164	1,228
	[lifestyle = 1]	0^b	.	.	0	.	.	.	.
Medium	Intercept	20,133	2,329	74,743	1	,000
	age	−,355	,040	79,904	1	,000	,701	,649	,758
	[gender = 0]	,306	,372	,674	1	,412	1,358	,654	2,817
	[gender = 1]	0^b	.	.	0	.	.	.	.
	[married = 0]	,612	,394	2,406	1	,121	1,843	,851	3,992
	[married = 1]	0^b	.	.	0	.	.	.	.
	[lifestyle = 0]	−,014	,382	,001	1	,972	,987	,466	2,088
	[lifestyle = 1]	0^b	.	.	0	.	.	.	.

^aThe reference category is: high

^bThis parameter is set to zero because it is redundant

The following conclusions are appropriate.

The unmarried subjects have a greater chance of QOL level 1 than the married ones (the b-value is positive here).

The higher the age, the less chance of having the low QOL levels 1 and 2 (the b-values (regression coefficients) are negative here). If you wish, you may also report the odds ratios (Exp (B) values) here.

6 Logit Loglinear Modeling

We will now perform a logit loglinear analysis. For analysis the statistical model Logit in the module Loglinear is required.

Command:

Analyze.... Loglinear....Logit....Dependent: enter “qol”....Factor(s): enter “gender, married, lifestyle”....Cell Covariate(s): enter: “age”....Model: Terms in Model: enter: “gender, married, lifestyle, age”....click Continue....click Options....mark Estimates....mark Adjusted residuals....mark normal probabilities for adjusted residuals....click Continue....click OK.

The table on page 306 shows the observed frequencies per cell, and the frequencies to be expected, if the predictors had no effect on the outcome.

Cell counts and residuals^a,b

				Observed		Expected
Gender	Married	Lifestyle	Qol	Count	%	Count	%	Residual	Standardized residual	Adjusted residual	Deviance
Male	Unmarried	Inactive	Low	7	23,3 %	9,111	30,4 %	−2,111	−,838	−1,125	−1,921
			Medium	16	53,3 %	14,124	47,1 %	1,876	,686	,888	1,998
			High	7	23,3 %	6,765	22,6 %	,235	,103	,127	,691
		Active	Low	29	61,7 %	25,840	55,0 %	3,160	,927	2,018	2,587
			Medium	5	10,6 %	10,087	21,5 %	−5,087	−1,807	−2,933	−2,649
			High	13	27,7 %	11,074	23,6 %	1,926	,662	2,019	2,042
	Married	Inactive	Low	9	11,0 %	10,636	13,0 %	−1,636	−,538	−,826	−1,734
			Medium	41	50,0 %	43,454	53,0 %	−2,454	−,543	−1,062	−2,183
			High	32	39,0 %	27,910	34,0 %	4,090	,953	2,006	2,958
		Active	Low	15	23,8 %	14,413	22,9 %	,587	,176	,754	1,094
			Medium	27	42,9 %	21,336	33,9 %	5,664	1,508	2,761	3,566
			High	21	33,3 %	27,251	43,3 %	−6,251	−1,590	−2,868	−3,308
Female	Unmarried	Inactive	Low	12	26,1 %	11,119	24,2 %	,881	,303	,627	1,353
			Medium	26	56,5 %	22,991	50,0 %	3,009	,887	1,601	2,529
			High	8	17,4 %	11,890	25,8 %	−3,890	−1,310	−1,994	−2,518
		Active	Low	18	54,5 %	19,930	60,4 %	−1,930	−,687	−,978	−1,915
			Medium	6	18,2 %	5,799	17,6 %	,201	,092	,138	,639
			High	9	27,3 %	7,271	22,0 %	1,729	,726	1,064	1,959
	Married	Inactive	Low	15	18,5 %	12,134	15,0 %	2,866	,892	1,670	2,522
			Medium	27	33,3 %	29,432	36,3 %	−2,432	−,562	−1,781	−2,158
			High	39	48,1 %	39,434	48,7 %	−.434	−,097	−,358	−,929
		Active	Low	16	25,4 %	17,817	28,3 %	−1,817	−,508	−1,123	−1,855
			Medium	24	38,1 %	24,779	39,3 %	−,779	−,201	−,882	−1,238
			High	23	36,5 %	20,404	32,4 %	2,596	,699	1,407	2,347

^aModel: Multinomial Logit

^bDesign: Constant + qol + qol* gender + qol* married + qol* lifestyle + qol* age

The underneath table shows the results of the statistical tests of the data.

Parameter estimates^a,b

Parameter		Estimate	Std. error	Z	Sig.	95 % confidence interval
Parameter		Estimate	Std. error	Z	Sig.	Lower bound	Upper bound
Constant	[gender = 0]* [married = 0] * [lifestyle = 0]	−7,402^c
	[gender = 0]* [married = 0]* [lifestyle = 1]	−7,409^c
	[gender = 0]* [married = 1]* [lifestyle = 0]	−6,088^c
	[gender = 0]* [married = 1]* [lifestyle = 1]	−6,349^c
	[gender = 1][married = 0] [lifestyle = 0]	−6,825^c
	[gender = 1]* [married = 0]* [lifestyle = 1]	−7,406^c
	[gender = 1]* [married = 1]* [lifestyle = 0]	−5,960^c
	[gender = 1][married = 1] [lifestyle = 1]	−6,567^c
[qol = 1]		5,332	8,845	,603	,547	−12,004	22,667
[qol = 2]		4,280	10,073	,425	,671	−15,463	24,022
[qol = 3]		0^d	.	.	.	.	.
[qol = 1]* [gender = 0]		,389	,360	1,079	,280	−,317	1,095
[qol = 1]* [gender = 1]		0^d	.	.	.	.	.
[qol = 2]* [gender = 0]		−,140	,265	−,528	,597	−,660	,380
[qol = 2]* [gender = 1]		0^d	.	.	.	.	.
[qol = 3]* [gender = 0]		0^d	.	.	.	.	.
[qol = 3]* [gender = 1]		0^d	.	.	.	.	.
[qol = 1]* [married = 0]		1,132	,283	4,001	,000	,578	1,687
[qol = 1]* [married = 1]		0^d	.	.	.	.	.
[qol = 2]* [married = 0]		−,078	,294	−,267	,790	−,655	,498
[qol = 2]* [married = 1]		0^d	.	.	.	.	.
[qol = 3]* [married = 0]		0^d	.	.	.	.	.
[qol = 3]* [married = 1]		0^d	.	.	.	.	.
[qol = 1]* [lifestyle = 0]		−1,004	,311	−3,229	,001	−1,613	−,394
[qol = 1]* [lifestyle = 1]		0^d	.	.	.	.	.
[qol = 2] * [lifestyle = 0]		,016	,271	,059	,953	−,515	,547
[qol = 2]* [lifestyle = 1]		0^d	.	.	.	.	.
[qol = 3]* [lifestyle = 0]		0^d	.	.	.	.	.
[qol = 3]* [lifestyle = 1]		0^d	.	.	.	.	.
[qol = 1]* age		,116	,074	1,561	,119	−,030	,261
[qol = 2]* age		,114	,054	2,115	,034	,008	,219
[qol = 3]* age		,149	,138	1,075	,282	−,122	,419

^aModel: Multinomial Logit

^bDesign: Constant + qol + qol* gender + qol* married + qol* lifestyle + qol* age

^cConstants are not parameters under the multinomial assumption. Therefore, their standard errors are not calculated

^dThis parameter is set to zero because it is redundant

The following conclusions are appropriate.

The unmarried subjects have a greater chance of QOL 1 (low QOL) than their married counterparts.

The inactive lifestyle subjects have a greater chance of QOL 1 (low QOL) than their adequate-lifestyle counterparts.

The higher the age the more chance of QOL 2 (medium level QOL), which is neither very good nor very bad, nut rather in-between (as you would expect).

We may conclude that the two procedures produce similar results, but the latter method provides some additional information about the lifestyle.

7 Conclusion

Multinomial regression is adequate for identifying the main predictors of outcome categories, like levels of injury or quality of life. An alternative approach is logit loglinear modeling. The latter method does not use continuous predictors on a case by case basis, but rather the weighted means of subgroups formed with the help of the discrete predictors. This approach allowed for relevant additional conclusions in the example given.

8 Note

More background, theoretical and mathematical information of polytomous/multinomial regression is given in the Chap.44. More information of loglinear modeling is in the Chaps. 24 and 52.

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