Question

1. Choose a data set of your own:?Response or dependent variable (Y)?At least 3 or more independent variables (X1, X2, X3, ... etc.) that you believe has an influence on Y.?At least 40 observations or data points?If there are categorical variables, model them appropriately2. Fit a multiple regression model. ?Interpret the model equation?Are all the chosen variables significant? Discuss.?Check for model assumptions and make appropriate comments.?How good is the model? Comment on R2 , R , se, F-value etc and discuss.2a3. Refine the model.?Does the addition of interaction terms create a better model??Can any variable be eliminated??Does stepwise regression, forward selection, or backward elimination produce the same model? Comment and substantiate.4. Forecast and predict using the chosen model.?Find a 95% confidence interval for the mean value of Y for given value of X’s?Find a 95% prediction interval for a particular value of Y for a given value of X’s?Comment and discuss the two above intervals?Are there any influential observations? Comment and discuss.5. Conclusions?Are you comfortable with the model??If you had to re-build the model, what changes would you consider?6. Appendix?Include copy of the data set?Include figures, tables, outputs of analyses using software. Be sure to clearly label each so that you may refer to them in your write-up.

Answer 1

In an example we want to test whether the different variables affect the price of the residential home. The previous data of 150 residential apartment is shown below,

Home	Price	Home Size	Lot Size	Rooms	Bathrooms
1	102000	600	0.5	3	1
2	146300	1050	0.43	5	1.5
3	182000	1800	0.68	7	1.5
4	110500	922	0.3	5	1
5	171900	1950	0.75	8	2.5
6	154000	1783	0.22	8	1.5
7	147000	1008	0.5	6	1
8	195900	1840	1.16	8	2
9	183500	3700	1.1	10	3
10	156500	1092	0.26	6	1
11	152000	1950	0.5	7	1.5
12	170000	1403	0.5	6	2
13	253000	1680	14.37	8	2
14	129500	1000	0.49	4	1
15	241900	2310	0.46	8	2.5
16	151900	1300	0.78	6	1
17	199000	1930	3	9	3
18	186000	3000	0.5	11	2.5
19	153500	1362	0.4	7	2
20	166000	1750	0.5	7	2
21	224900	2080	1	8	2.5
22	158500	1344	0.94	6	2
23	332000	2130	11.91	8	1.5
24	172000	1500	0.41	7	1
25	176000	2400	0.4	7	2.5
26	210000	2272	0.41	9	2.5
27	156500	1050	1	5	1
28	169500	1610	0.45	8	1.5
29	154900	1248	0.22	7	1
30	163000	2000	0.5	8	2
31	140000	1450	0.3	6	2
32	148500	1248	0.25	7	1
33	224500	2544	0.28	9	2.5
34	299900	2500	0.92	8	3
35	199900	2858	0.79	9	3
36	220000	1745	0.58	7	2.5
37	233000	2653	1.8	9	3
38	174900	1450	0.3	7	1
39	124000	850	0.11	4	1
40	169900	1839	2.6	7	1.5
41	213000	2016	0.78	8	2.5
42	165000	1625	0.36	7	1.5
43	162000	2000	0.11	8	2
44	211500	2250	0.33	9	2.5
45	166000	1300	0.3	7	1
46	194000	1956	0.5	8	2.5
47	192000	2496	0.75	9	2.5
48	171000	1575	0.25	7	1.5
49	226800	1960	1.33	8	2.5
50	155000	1200	0.33	5	1
51	157500	1296	0.5	9	1
52	297000	1950	18.7	7	2.5
53	315000	2516	8.1	7	2.5
54	161000	1066	0.33	5	1
55	193500	2276	1	8	2.5
56	163000	1908	0.46	7	2
57	180000	1122	3.09	5	2
58	171000	3500	1	10	2.5
59	163000	1100	0.33	6	1
60	220000	2300	5.63	7	2.5
61	155900	1118	0.56	7	1.5
62	219900	2464	0.43	8	2.5
63	185000	2100	0.58	8	1.5
64	172500	1552	0.46	6	1.5
65	167900	1856	0.33	7	1.5
66	160000	1800	0.3	7	1.5
67	147000	1248	0.3	6	1
68	210500	2000	0.6	9	2.5
69	192500	1848	0.5	7	2.5
70	138000	1036	0.95	6	1
71	200000	2277	0.8	8	3
72	186000	2300	0.65	7	3
73	217000	2080	1.23	8	2.5
74	180000	1600	1.84	7	2
75	195000	2680	0.5	9	3
76	149000	1200	0.25	7	1
77	165500	1526	0.3	7	1.5
78	175900	1680	0.5	6	1.5
79	156000	1232	0.31	6	2
80	235406	2465	1.55	8	2.5
81	215500	2800	1.68	9	1.5
82	225000	2265	0.85	8	2.5
83	155000	1300	0.65	5	1
84	190000	1900	1	8	2.5
85	126000	864	0.32	4	1
86	172000	2000	0.75	9	1.5
87	175000	1800	0.66	8	2.5
88	181500	1900	0.75	7	2
89	180000	1564	0.33	6	2
90	295000	2400	2	7	2
91	146000	1100	1.1	6	1
92	165000	1800	1	8	2.5
93	159000	1200	0.33	6	1
94	138500	1540	0.18	7	2
95	194900	1980	0.7	8	2.5
96	140000	1289	0.25	6	1
97	184000	1800	0.68	7	2
98	164000	1502	0.35	7	1.5
99	190000	2025	1.1	7	2
100	250000	3000	1.15	10	3.5
101	156500	1500	0.5	7	1.5
102	156500	1600	0.26	8	1.5
103	188000	1500	0.54	5	2.5
104	202000	2100	1	8	2.5
105	245000	2100	0.5	8	2.5
106	171900	1632	3	6	3
107	119900	1660	0.21	7	1
108	159900	1070	1.69	5	1
109	165000	1400	0.35	6	2
110	165000	1800	0.5	7	2
111	152500	1100	0.37	7	1
112	265000	3150	0.3	11	4
113	164500	2000	0.7	8	1
114	156500	1700	0.3	8	2
115	210000	1800	1.52	8	2.5
116	157500	1850	0.26	9	2
117	195000	2320	0.4	8	2.5
118	127000	1300	0.37	5	1
119	130000	1338	0.12	6	1
120	238000	2288	1.2	8	2.5
121	212000	2400	0.5	8	2.5
122	205000	2400	0.7	8	3
123	174900	1900	0.44	6	2
124	207000	2010	0.68	8	1.5
125	261750	2981	1.3	10	3.5
126	195000	1725	1.53	8	2.5
127	108000	821	2.3	4	1
128	209000	3060	0.75	8	2
129	115000	875	0.26	5	1
130	190000	1760	0.05	7	2
131	171000	2000	0.65	7	1
132	215000	2600	0.75	8	2
133	143500	1624	1.8	7	1.5
134	220000	2473	1.25	9	2.5
135	137000	1100	0.17	5	1
136	247000	3100	0.54	10	3.5
137	224500	2300	0.91	8	2.5
138	182000	1450	0.3	6	1.5
139	240000	2100	0.5	8	2.5
140	170000	1650	0.5	8	2.5
141	150500	1600	0.4	6	2
142	209900	2790	0.75	13	2.5
143	182500	1786	0.3	8	2
144	189000	1728	0.5	8	1.5
145	198500	1900	1.06	7	2.5
146	128000	1165	0.12	6	1
147	147500	1300	0.29	6	1
148	145000	1080	0.31	5	1
149	305000	2820	1	9	2.5
150	220000	2100	1.3	8	1.5

The regression analysis is done in excel by using following steps,

Step 1: Write the data value in excel. the sceenshot is shown below,

Step 2: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

Step 3: Select Input Y Range: Price column, Input X Range: All the independent variables and tick on Confidence level = 95%. The screenshot is shown below,

The Result is obtained. The screenshot is shown below,

The multiple regression equation is,

Y = 85205 + 30.88547x1 + 7535.056X2 823 . 2338x3 + 14722.24X4

The R-square value is 0.682428 which means model fits the 68.2428% of the data.

The F statistic value = 77.89748 < Significance F. There is a significant effect of independent variables on price.

The P-values of each indepent variables are,

	P-value
Home Size	0.000<0.05
Lot Size	0.000<0.05
Rooms	0.724>0.05
Bathrooms	0.0008<0.05

The P-value is significant for all the variable except Rooms. Variable rooms is insignificant variable here. Hence this variable needs to be eliminated from the regression equation.

Let say the variable Home size and Bathroom has an interaction present. Now making the regression model by adding the new variable Home Size * Bathroom, Home Size * Lot Size and Lot Size and bathroom. The screenshot of result summary is shown below,

There is a significant effect of Home size * lot size and Lot size * Bathroom while Home size * Bathroom do not hava significant interaction effect.

The adjusted R square value is 0.7091 which is 70.91%.

By removing the insignificant variable Home size * Bathroom. The screenshot of summary output is shown below

R square value = 0.708452

(Note: Use forward stepwise method to add variable one by one)

The model will give the same result to whether select forward stepwise method or backward stepwise method.

Forcast of model using model with interaction

For Home size 800, Lot size = 0.75, bathroom = 1.5

Using the value in last model regression line,

$\widehat{Y}=96913.13+17.29X_1-2203.03X_2+24400.65X_3-6248.55X_2X_3+11.99X_1X_2$