Example 24.2: Multiple Correspondence Analysis of Cars and Their Owners

The CORRESP Procedure

Example 24.2: Multiple Correspondence Analysis of Cars and Their Owners

In this example, PROC CORRESP creates a Burt table from categorical data and performs a multiple correspondence analysis. The data are from a sample of individuals who were asked to provide information about themselves and their cars. The questions included origin of the car (American, Japanese, European), size of car (Small, Medium, Large), type of car (Family, Sporty, Work Vehicle), home ownership (Owns, Rents), marital/family status (single, married, single and living with children, and married living with children), and sex (Male, Female).

The data are read and formats assigned in a previous step, displayed in Example 24.1. The variables used in this example are Origin, Size, Type, Income, Home, Marital, and Sex. MCA specifies multiple correspondence analysis, OBSERVED displays the Burt table, and the OUTC= option creates an output coordinate data set. The TABLES statement with only a single variable list and no comma creates the Burt table. The %PLOTIT macro is used to plot the results with vertical and horizontal reference lines. The data used to produce Output 24.2.1 and Output 24.2.2 can be found in Example 24.1.

   title 'MCA of Car Owners and Car Attributes';

   *---Perform Multiple Correspondence Analysis---;
   proc corresp mca observed data=Cars outc=Coor;
      tables Origin Size Type Income Home Marital Sex;
   run;

   *---Plot the Multiple Correspondence Analysis Results---;
   %plotit(data=Coor, datatype=corresp, HREF=0, vref=0)

Output 24.2.1: Multiple Correspondence Analysis of a Burt Table

MCA of Car Owners and Car Attributes

The CORRESP Procedure

Burt Table
	American	European	Japanese	Large	Medium	Small	Family	Sporty	Work	1 Income	2 Incomes	Own	Rent	Married	Married with Kids	Single	Single with Kids	Female	Male
American	125	0	0	36	60	29	81	24	20	58	67	93	32	37	50	32	6	58	67
European	0	44	0	4	20	20	17	23	4	18	26	38	6	13	15	15	1	21	23
Japanese	0	0	165	2	61	102	76	59	30	74	91	111	54	51	44	62	8	70	95
Large	36	4	2	42	0	0	30	1	11	20	22	35	7	9	21	11	1	17	25
Medium	60	20	61	0	141	0	89	39	13	57	84	106	35	42	51	40	8	70	71
Small	29	20	102	0	0	151	55	66	30	73	78	101	50	50	37	58	6	62	89
Family	81	17	76	30	89	55	174	0	0	69	105	130	44	50	79	35	10	83	91
Sporty	24	23	59	1	39	66	0	106	0	55	51	71	35	35	12	57	2	44	62
Work	20	4	30	11	13	30	0	0	54	26	28	41	13	16	18	17	3	22	32
1 Income	58	18	74	20	57	73	69	55	26	150	0	80	70	10	27	99	14	47	103
2 Incomes	67	26	91	22	84	78	105	51	28	0	184	162	22	91	82	10	1	102	82
Own	93	38	111	35	106	101	130	71	41	80	162	242	0	76	106	52	8	114	128
Rent	32	6	54	7	35	50	44	35	13	70	22	0	92	25	3	57	7	35	57
Married	37	13	51	9	42	50	50	35	16	10	91	76	25	101	0	0	0	53	48
Married with Kids	50	15	44	21	51	37	79	12	18	27	82	106	3	0	109	0	0	48	61
Single	32	15	62	11	40	58	35	57	17	99	10	52	57	0	0	109	0	35	74
Single with Kids	6	1	8	1	8	6	10	2	3	14	1	8	7	0	0	0	15	13	2
Female	58	21	70	17	70	62	83	44	22	47	102	114	35	53	48	35	13	149	0
Male	67	23	95	25	71	89	91	62	32	103	82	128	57	48	61	74	2	0	185

MCA of Car Owners and Car Attributes

The CORRESP Procedure

Inertia and Chi-Square Decomposition
Singular Value	Principal Inertia	Chi- Square	Percent	Cumulative Percent	4 8 12 16 20 ----+----+----+----+----+---
0.56934	0.32415	970.77	18.91	18.91	************************
0.48352	0.23380	700.17	13.64	32.55	*****************
0.42716	0.18247	546.45	10.64	43.19	*************
0.41215	0.16987	508.73	9.91	53.10	************
0.38773	0.15033	450.22	8.77	61.87	***********
0.38520	0.14838	444.35	8.66	70.52	***********
0.34066	0.11605	347.55	6.77	77.29	********
0.32983	0.10879	325.79	6.35	83.64	********
0.31517	0.09933	297.47	5.79	89.43	*******
0.28069	0.07879	235.95	4.60	94.03	******
0.26115	0.06820	204.24	3.98	98.01	*****
0.18477	0.03414	102.24	1.99	100.00	**
Total	1.71429	5133.92	100.00
Degrees of Freedom = 324

MCA of Car Owners and Car Attributes

The CORRESP Procedure

Column Coordinates
	Dim1	Dim2
American	-0.4035	0.8129
European	-0.0568	-0.5552
Japanese	0.3208	-0.4678
Large	-0.6949	1.5666
Medium	-0.2562	0.0965
Small	0.4326	-0.5258
Family	-0.4201	0.3602
Sporty	0.6604	-0.6696
Work	0.0575	0.1539
1 Income	0.8251	0.5472
2 Incomes	-0.6727	-0.4461
Own	-0.3887	-0.0943
Rent	1.0225	0.2480
Married	-0.4169	-0.7954
Married with Kids	-0.8200	0.3237
Single	1.1461	0.2930
Single with Kids	0.4373	0.8736
Female	-0.3365	-0.2057
Male	0.2710	0.1656

Summary Statistics for the Column Points
	Quality	Mass	Inertia
American	0.4925	0.0535	0.0521
European	0.0473	0.0188	0.0724
Japanese	0.3141	0.0706	0.0422
Large	0.4224	0.0180	0.0729
Medium	0.0548	0.0603	0.0482
Small	0.3825	0.0646	0.0457
Family	0.3330	0.0744	0.0399
Sporty	0.4112	0.0453	0.0569
Work	0.0052	0.0231	0.0699
1 Income	0.7991	0.0642	0.0459
2 Incomes	0.7991	0.0787	0.0374
Own	0.4208	0.1035	0.0230
Rent	0.4208	0.0393	0.0604
Married	0.3496	0.0432	0.0581
Married with Kids	0.3765	0.0466	0.0561
Single	0.6780	0.0466	0.0561
Single with Kids	0.0449	0.0064	0.0796
Female	0.1253	0.0637	0.0462
Male	0.1253	0.0791	0.0372

MCA of Car Owners and Car Attributes

The CORRESP Procedure

Partial Contributions to Inertia for the Column Points
	Dim1	Dim2
American	0.0268	0.1511
European	0.0002	0.0248
Japanese	0.0224	0.0660
Large	0.0268	0.1886
Medium	0.0122	0.0024
Small	0.0373	0.0764
Family	0.0405	0.0413
Sporty	0.0610	0.0870
Work	0.0002	0.0023
1 Income	0.1348	0.0822
2 Incomes	0.1099	0.0670
Own	0.0482	0.0039
Rent	0.1269	0.0103
Married	0.0232	0.1169
Married with Kids	0.0967	0.0209
Single	0.1889	0.0171
Single with Kids	0.0038	0.0209
Female	0.0223	0.0115
Male	0.0179	0.0093

MCA of Car Owners and Car Attributes

The CORRESP Procedure

Indices of the Coordinates that Contribute Most to Inertia for the Column Points
	Dim1	Dim2	Best
American	0	2	2
European	0	0	2
Japanese	0	2	2
Large	0	2	2
Medium	0	0	1
Small	0	2	2
Family	2	0	2
Sporty	2	2	2
Work	0	0	2
1 Income	1	1	1
2 Incomes	1	1	1
Own	1	0	1
Rent	1	0	1
Married	0	2	2
Married with Kids	1	0	1
Single	1	0	1
Single with Kids	0	0	2
Female	0	0	1
Male	0	0	1

Squared Cosines for the Column Points
	Dim1	Dim2
American	0.0974	0.3952
European	0.0005	0.0468
Japanese	0.1005	0.2136
Large	0.0695	0.3530
Medium	0.0480	0.0068
Small	0.1544	0.2281
Family	0.1919	0.1411
Sporty	0.2027	0.2085
Work	0.0006	0.0046
1 Income	0.5550	0.2441
2 Incomes	0.5550	0.2441
Own	0.3975	0.0234
Rent	0.3975	0.0234
Married	0.0753	0.2742
Married with Kids	0.3258	0.0508
Single	0.6364	0.0416
Single with Kids	0.0090	0.0359
Female	0.0912	0.0341
Male	0.0912	0.0341

Multiple correspondence analysis locates all the categories in a Euclidean space. The first two dimensions of this space are plotted to examine the associations among the categories. The top-right quadrant of the plot shows that the categories single, single with kids, 1 income, and renting a home are associated. Proceeding clockwise, the categories sporty, small, and Japanese are associated. The bottom-left quadrant shows the association between being married, owning your own home, and having two incomes. Having children is associated with owning a large American family car. Such information could be used in market research to identify target audiences for advertisements.

This interpretation is based on points found in approximately the same direction from the origin and in approximately the same region of the space. Distances between points do not have a straightforward interpretation in multiple correspondence analysis. The geometry of multiple correspondence analysis is not a simple generalization of the geometry of simple correspondence analysis (Greenacre and Hastie 1987; Greenacre 1988).

Output 24.2.2: Plot of Multiple Correspondence Analysis of a Burt Table

If you want to perform a multiple correspondence analysis and get scores for the individuals, you can specify the BINARY option to analyze the binary table.

   title 'Car Owners and Car Attributes';
   title2 'Binary Table';

   *---Perform Multiple Correspondence Analysis---;
   proc corresp data=Cars binary;
      ods select RowCoors;
      tables Origin Size Type Income Home Marital Sex;
   run;

Output 24.2.3: Correspondence Analysis of a Binary Table

Car Owners and Car Attributes

Binary Table

The CORRESP Procedure

Row Coordinates
	Dim1	Dim2
1	-0.4093	1.0878
2	0.8198	-0.2221
3	-0.2193	-0.5328
4	0.4382	1.1799
5	-0.6750	0.3600
6	-0.1778	0.1441
7	-0.9375	0.6846
8	-0.7405	-0.1539
9	-0.3027	-0.2749
10	-0.7263	-0.0803
11	-0.2965	-0.6420
12	0.5522	-0.0640
13	-0.4552	-0.3846
14	0.8198	-0.2221
15	-0.1524	-0.1371
16	-0.4093	1.0878
17	0.6381	0.1563
18	0.3140	0.5835
19	0.0554	0.0913
20	-0.5880	-0.0442
21	-0.0176	-0.5414
22	-0.4933	-0.0183
23	-0.4093	1.0878
24	0.5726	-0.3576
25	-0.7405	-0.1539
26	-0.3682	-0.7888
27	1.1739	-0.1209
28	-0.4552	-0.3846
29	-0.1006	-0.9468
30	-0.1299	-0.4587
31	-0.5446	-0.4587
32	0.9922	0.2574
33	-0.6393	-0.4845
34	0.1752	0.0303
35	-0.6652	0.7334
36	1.0011	0.0629
37	0.3958	0.6358
38	-0.7405	-0.1539
39	-0.0065	0.4087
40	-0.1006	-0.9468
41	-0.4552	-0.3846
42	0.4856	0.0466
43	0.8486	-0.0468
44	0.0549	0.7546
45	0.9028	0.1833
46	-0.6457	-0.1280
47	0.4652	0.3402
48	0.0804	0.0045
49	-0.0429	-0.8629
50	-0.4729	-0.3119
51	-0.2992	0.6535
52	-0.6457	-0.1280
53	-0.3204	-0.2022
54	0.3759	0.2661
55	-0.0469	-0.0534
56	-0.1210	-0.6532
57	0.4382	1.1799
58	-0.0429	-0.8629
59	-0.9375	0.6846
60	-0.3081	0.7572
61	0.5161	-0.0885
62	-0.3204	-0.2022
63	-0.2992	0.6535
64	-0.1175	0.2751
65	-0.2193	-0.5328
66	-0.5880	-0.0442
67	-0.7851	0.7944
68	-0.9375	0.6846
69	1.0226	0.1224
70	-0.1954	-0.9726
71	1.1739	-0.1209
72	0.5580	1.1189
73	-0.3722	0.0208
74	0.1384	0.5039
75	0.5161	-0.0885
76	-0.2122	0.2493
77	-0.3531	-0.3728
78	0.6806	-0.1119
79	-0.5022	0.1762
80	-0.3204	-0.2022
81	-0.1883	0.6962
82	0.2179	0.8073
83	-0.8275	0.2503
84	0.2535	-0.8457
85	-0.3531	-0.3728
86	0.7300	0.3672
87	-0.5535	-0.2642
88	-0.3921	-0.3490
89	-0.9375	0.6846
90	0.1565	-0.2394
91	0.0818	-0.3088
92	1.0215	-0.2306
93	1.0226	0.1224
94	-0.2812	-0.2154
95	0.0518	-0.8371
96	-0.3204	-0.2022
97	-0.5348	-0.0853
98	-0.6296	-0.1111
99	-0.1210	-0.6532
100	-0.3717	-0.6426
101	-0.5739	0.0294
102	-0.4933	-0.0183
103	-0.1905	-0.3576
104	-0.2222	-0.3226
105	-0.3141	-0.5587
106	0.5487	0.0822
107	-0.5563	-0.0539
108	0.2242	-0.3576
109	0.1565	0.4614
110	-0.1006	-0.9468
111	0.7539	-0.0726
112	0.8193	0.4413
113	1.1739	-0.1209
114	0.6450	0.7147
115	0.8498	0.3062
116	0.1764	-0.3575
117	0.0818	-0.3088
118	-0.3141	-0.5587
119	0.4652	0.3402
120	0.7504	0.0736
121	0.7504	0.0736
122	0.3128	0.2305
123	-0.5446	-0.4587
124	-0.4010	-0.1545
125	-0.2992	0.6535
126	-0.5535	-0.2642
127	0.8198	-0.2221
128	-0.1264	1.0796
129	0.2219	-0.0022
130	0.1187	-0.4414
131	0.6470	-0.0382
132	-0.1006	-0.9468
133	-0.8275	0.2503
134	-0.6393	-0.4845
135	0.2039	1.0178
136	-0.7851	0.7944
137	0.2546	-0.4927
138	-0.3531	-0.3728
139	-0.6546	0.0665
140	-0.5446	-0.4587
141	-0.7165	0.2931
142	0.2179	0.8073
143	-0.7263	-0.0803
144	0.2916	-0.6252
145	-0.8505	0.2805
146	0.2712	-0.3317
147	-0.4093	1.0878
148	0.6674	-0.3318
149	-0.3204	-0.2022
150	0.8198	-0.2221
151	0.4440	-0.5155
152	0.2634	-0.2738
153	-0.3734	-0.0793
154	-0.6750	0.3600
155	0.7250	0.3548
156	-0.3746	-0.4323
157	-0.9375	0.6846
158	-0.5446	-0.4587
159	1.0011	0.0629
160	-0.1954	-0.9726
161	-0.2006	-0.2631
162	-0.9375	0.6846
163	-0.8275	0.2503
164	-0.5348	0.0055
165	-0.4552	-0.3846
166	-0.4093	1.0878
167	0.2179	0.8073
168	-0.8275	0.2503
169	1.0215	-0.2306
170	-0.7851	0.7944
171	-0.7263	-0.0803
172	0.7981	-0.4144
173	0.1035	-0.1165
174	-0.4729	-0.3119
175	0.0417	0.5347
176	-0.3734	-0.0793
177	-0.7851	0.7944
178	0.3963	-0.0275
179	0.8198	-0.2221
180	-0.3717	-0.6426
181	-0.8177	0.6237
182	-0.2275	0.7095
183	-0.3531	-0.3728
184	-0.0429	-0.8629
185	0.0554	0.0913
186	-0.7263	-0.0803
187	0.1764	0.1304
188	0.7251	-0.2479
189	-0.5022	0.1762
190	0.8198	-0.2221
191	-0.5446	-0.4587
192	1.0226	0.1224
193	0.3997	0.4290
194	0.0518	-0.8371
195	-0.1361	0.7062
196	1.0215	-0.2306
197	0.5487	0.0822
198	1.0011	0.0629
199	-0.2992	0.6535
200	1.1739	-0.1209
201	-0.2916	-0.0269
202	-0.4933	-0.0183
203	-0.6457	-0.1280
204	-0.7263	-0.0803
205	-0.2006	-0.2631
206	0.2234	0.1563
207	0.1560	0.4240
208	0.8409	0.5007
209	0.4945	-0.1479
210	0.1287	0.1305
211	1.1739	-0.1209
212	0.1348	-0.4317
213	-0.1006	-0.9468
214	-0.0163	-0.0555
215	-0.5739	0.0294
216	1.1739	-0.1209
217	0.7504	0.0736
218	-0.4933	-0.0183
219	0.5482	0.7455
220	-0.1264	0.4696
221	-0.0493	-0.5064
222	0.6000	-0.0642
223	-0.4729	-0.3119
224	0.8436	-0.0591
225	-0.6750	0.3600
226	-0.5022	0.1762
227	1.0226	0.1224
228	0.4382	1.1799
229	-0.8275	0.2503
230	-0.3717	-0.6426
231	0.5775	0.2575
232	0.7300	0.3672
233	1.0215	-0.2306
234	1.0215	-0.2306
235	0.7251	-0.2479
236	0.4059	-0.7360
237	0.9028	0.1833
238	-0.8275	0.2503
239	-0.6546	0.0665
240	0.6669	0.3316
241	-0.1006	-0.9468
242	0.4272	0.1197
243	-0.8364	0.3540
244	-0.6750	0.3600
245	0.8198	-0.2221
246	0.0841	1.0787
247	-0.1794	0.5925
248	-0.2193	-0.5328
249	0.0841	1.0787
250	-0.1175	0.2751
251	-0.0206	0.4810
252	-0.1441	-0.5323
253	0.5482	0.7455
254	0.8193	0.4413
255	-0.6457	-0.1280
256	-0.6065	-0.1413
257	-0.1441	-0.5323
258	-0.4552	-0.3846
259	0.8080	0.1575
260	0.6685	0.0213
261	0.9267	-0.2565
262	0.8397	0.1477
263	-0.4336	-0.3251
264	-0.3298	0.5648
265	0.4945	-0.1479
266	-0.2018	-0.6162
267	-0.2519	-0.7035
268	0.5522	-0.0640
269	-0.2222	-0.3226
270	-0.4093	1.0878
271	-0.5446	-0.4587
272	0.5725	0.2451
273	-0.4823	0.4551
274	0.4059	-0.7360
275	-0.0380	-0.2479
276	0.8198	-0.2221
277	-0.8364	0.3540
278	-0.2519	-0.7035
279	-0.6457	-0.1280
280	0.1764	-0.3575
281	0.2535	-0.8457
282	-0.5259	-0.1890
283	-0.8275	0.2503
284	0.2184	0.1440
285	0.2184	0.1440
286	-0.8275	0.2503
287	0.0518	-0.8371
288	-0.5348	0.0055
289	-0.2519	-0.7035
290	-0.6457	-0.1280
291	-0.2193	-0.5328
292	0.0841	1.0787
293	-0.5552	0.2991
294	0.1348	-0.4317
295	-0.2158	-0.6791
296	0.3265	-0.2130
297	-0.2965	-0.6420
298	-0.1210	-0.6532
299	-0.3921	-0.3490
300	-0.0469	-0.0534
301	0.1941	0.6444
302	0.1654	0.2669
303	0.5726	-0.3576
304	-0.4540	-0.0316
305	0.7300	0.3672
306	0.0554	0.0913
307	-0.6457	-0.1280
308	-0.8275	0.2503
309	0.3759	0.2661
310	0.1729	-0.2113
311	0.5482	0.7455
312	-0.5880	-0.0442
313	-0.4729	-0.3119
314	-0.2122	0.2493
315	0.9028	0.1833
316	1.0226	0.1224
317	0.4868	0.3996
318	-0.1210	-0.6532
319	-0.0995	-0.5938
320	1.0226	0.1224
321	-0.1905	-0.3576
322	0.8198	-0.2221
323	-0.7405	-0.1539
324	-0.5535	-0.2642
325	0.9028	0.1833
326	0.6669	0.3316
327	0.8397	0.1477
328	-0.9375	0.6846
329	0.0518	-0.8371
330	-0.2992	0.6535
331	1.0791	-0.1468
332	0.3958	0.6358
333	-0.1905	-0.3576
334	0.5482	0.7455

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