Question

1．Understand, explain and apply regression theory

2. Choose or collect a data file, use regression theory to set up a model, calculate parameters value, get the regression model, analysis the meaning of model, including R square, F-test, t-test, explain the relation between dependent variable and independent variables. During the analysis, you need to represent the chart, correlation, regression output table. You’d better choose the multiple regression model, preferably one that includes dummy variables.

3. You also need to send me the operation in excel, the excel file.

Answer 1

Answers:

Ans 1)

Multiple Linear Regression Theory:

Regression analysis is a powerful statistical method that allows you to examine the relationship between two or more variables of interest. While there are many types of regression analysis, at their core they all examine the influence of one or more independent variables on a dependent variable

Multiple regression is an extension of simple linear regression. It is used when we want to predict the value of a variable based on the value of two or more other variables. The variable we want to predict is called the dependent variable (or sometimes, the outcome, target, or criterion variable).

Ans 2)

Collected Data: To compare the smokers and non-smokers

Dependent variable = Risk

Independent variable = Age, Pressure, Smoker

Categorical variable or dummy variable = Smoker, we assign the Yes = 1 and No = 0

=IF(E4="Yes",1,0)

Risk	Age	Pressure	Smoker		Risk(Y)	Age (X1)	Pressure(X2)	Smoker (X3)
12	57	152	No		12	57	152	0
24	67	163	No		24	67	163	0
13	58	155	No		13	58	155	0
56	86	177	Yes		56	86	177	1
28	59	196	No		28	59	196	0
51	76	189	Yes		51	76	189	1
18	56	155	Yes		18	56	155	1
31	78	120	No		31	78	120	0
37	80	135	Yes		37	80	135	1
15	78	98	No		15	78	98	0
22	71	152	No		22	71	152	0
36	70	173	Yes		36	70	173	1
15	67	135	Yes		15	67	135	1
48	77	209	Yes		48	77	209	1
15	60	199	No		15	60	199	0
36	82	119	Yes		36	82	119	1
8	66	166	No		8	66	166	0
34	80	125	Yes		34	80	125	1
3	62	117	No		3	62	117	0
37	59	207	Yes		37	59	207	1

Regression Output:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.934605
R Square	0.873487
Adjusted R Square	0.849766
Standard Error	5.756575
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	3	3660.74	1220.247	36.82301	2.06E-07
Residual	16	530.2104	33.13815
Total	19	4190.95
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-91.7595	15.22276	-6.02778	1.76E-05	-124.03	-59.4887
Age (X1)	1.076741	0.165964	6.487814	7.49E-06	0.724914	1.428568
Pressure(X2)	0.251813	0.045226	5.567951	4.24E-05	0.15594	0.347687
Smoker (X3)	8.739871	3.000815	2.912499	0.010174	2.378427	15.10132
RESIDUAL OUTPUT					PROBABILITY OUTPUT
Observation	Predicted Risk(Y)	Residuals	Standard Residuals		Percentile	Risk(Y)
1	7.89039	4.10961	0.777953		2.5	3
2	21.42775	2.572252	0.48693		7.5	8
3	9.722571	3.277429	0.62042		12.5	12
4	54.15109	1.848912	0.350001		17.5	13
5	21.12366	6.876335	1.301696		22.5	15
6	46.40544	4.594561	0.869754		27.5	15
7	16.30896	1.69104	0.320115		32.5	15
8	22.44392	8.556079	1.619674		37.5	18
9	37.11448	-0.11448	-0.02167		42.5	22
10	16.90402	-1.90402	-0.36043		47.5	24
11	22.96476	-0.96476	-0.18263		52.5	28
12	35.91598	0.084023	0.015906		57.5	31
13	23.11684	-8.11684	-1.53653		62.5	34
14	52.51845	-4.51845	-0.85535		67.5	36
15	22.95585	-7.95585	-1.50605		72.5	36
16	35.23894	0.761058	0.144069		77.5	37
17	21.10645	-13.1064	-2.48106		82.5	37
18	34.59634	-0.59634	-0.11289		87.5	48
19	4.460623	-1.46062	-0.2765		92.5	51
20	32.63348	4.366516	0.826585		97.5	56

Regression Model:

Predicted risk = -91.759 + 1.076 x Age + 0.251 x Pressure + 8.739 x Smoker

Interpretations:

2) All three independent variable age, pressure, and smoker are statistically significant because all three variables have the p-value is less than 0.05.

3)  r is always between -1 and 1 inclusive. The R-squared value, denoted by R ², is the square of the correlation. It measures the proportion of variation in the dependent variable that can be attributed to the independent variable. Correlation r = 0.9; R=squared = 0.87. Small positive linear association

Graphs:

1) Residual Plots:

2) Line Fit Plot:

3) Normal Probability Plot:

Ans 3).

Age (X1) Residual Plot 10 0 Residuals 20 40 60 80 100 -10 -20 Age (X1)

Pressure(X2) Residual Plot 10 0 Residuals 0 50 100 150 200 250 -10 -20 Pressure(X2)

Smoker (X3) Residual Plot 10 Residuals 0.2 0.4 0.6 0.8 1.2 -10 -20 Smoker (X3)

Age (X1) Line Fit Plot 60 40 Risk(Y) 20 Risk(Y) Predicted Risk(Y) 0 0 100 50 Age (X1)

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Smoker (X3) Line Fit Plot 60 40 Risk(Y) 20 Risk(Y) Predicted Risk(Y) 0 0 0.5 1 1.5 Smoker (X3)

Normal Probability Plot 60 40 Risk(Y) 20 0 0 20 100 120 40 60 80 Sample Percentile

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