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REGRESSION MODELS
By:
Ayush Sharma 09
Mickey Haldia 19
Prerna Makhijani 29
Sanoj George 39
Sushant Jaggi 49
Nitish Dorle 59
Example
Year Population on Farm (in millions)
1935 32.1
1940 30.5
1945 24.4
1950 23.0
1955 19.1
1960 15.6
1965 12.5
Scatter Plot
1930 1935 1940 1945 1950 1955 1960 1965 19700
5
10
15
20
25
30
35
Population(in millions)
Poplation(in millions)
Correlation Coefficient (r)
It is a measure of strength of the linear relationship between two variables and is calculated using the following formula:
Interpretation
After calculating we find r = -0.993
There is a strong negative correlation.
Coefficient of Determination Squaring the correlation coefficient (r) gives us
the percent variation in the y-variable that is described by the variation in the x-variable
To relate x and y, the Regression Equation is calculated using Least Squares technique.
Regression Equation: Y’ = a +bX Slope of the regression line:
To continue with the example We found r = -0.993. By squaring we get the
Coefficient of Determination (R^2) = 0.987
1930 1935 1940 1945 1950 1955 1960 1965 197010
15
20
25
30
35
f(x) = − 0.670714285714286 x + 1330.35R² = 0.986600589014608
Regression
Year
Po
pu
lati
on
on
Far
m (
in m
il-
lio
ns)
Interpretation
We conclude that 98.7% of the decrease in farm population can be explained by timeline progression.
Theoretically, population is a dependent variable (y-axis) and timeline is an independent variable (x-axis).
Assumptions of the Regression Model
The following assumptions are made about the errors:
a) The errors are independent
b) The errors are normally distributed
c) The errors have a mean of zero
d) The errors have a constant variance(regardless of the value of X)
Patterns of Indicating Errors
Error
X
Estimating the Variance
The error variance is measured by the MSE s2 = MSE= SSE
n-k-1
where n = number of observations in the sample
k = number of independent variables
Therefore the standard deviation will be
s = sqrt (MSE)
Multiple regression Analysis
More than one independent variable Y=β0+β1X1+β2X2+……+βkXk+ϵ
Where, Y=dependent variable(response variable)
Xi=ith independent variable(predictor variable or explanatory variable)
β0= intercept(value of Y when all Xi = 0)βi= coefficient of the ith independent variablek= number of independent variables ϵ= random error
To estimate the values of these coefficients, a sample is taken and the following equation is developed : Ῡ= b0+b1X1+b2X2+…….+bkXk
where, Ῡ= predicted value of Yb0= sample intercept (and is an estimate of
β0)bi= sample coefficient of ith variable(and is an estimate of βi)
Testing the Model for Significance
• MSE and co-efficient of determination (r2) does not provide a good measure of accuracy when the sample size is small
• In this case, it is necessary to test the model for significance
• Linear Model is given by,
Y=β0 + β1X + ε
Null Hypothesis :If β1 = 0, then there is no linear relationship between X and Y
Alternate Hypothesis : If β1 ≠ 0, then there is a linear relationship
Steps in Hypothesis Test for a Significant Regression Model
1. Specify null and alternative hypothesis.
2. Select the level of significance (α). Common values are between 0.01 and 0.05
3. Calculate the value of the test statistic using the formula:
F = MSE/MSE
4. Make a decision using one of the following methods:
a) Reject if Fcalculated > Ftable
b) Reject if p-value < α
Triple A Construction Example
Step 1:
H0 :β1 = 0, (no linear relationship between X and Y)
H1 :β1 ≠ 0, (linear relationship between X and Y)
Step 2
Select α = 0.05
Triple A Construction Example
Step 3: Calculate the value of the test statistic
MSR = SSR/k
= 15.6250/1
= 15.6250
F = MSR/MSE
= 15.6250/1.7188
= 9.09
Triple A Construction Example Step 4: Reject the null hypothesis if the test statistic is greater
than the F value from the table.
To find table value, we need :
Level of Significance (α) = 0.05
df1 = k = 1
df2 = n – k – 1 = 4
where k = number of independent variables
n = sample size
Using these values, we find
Ftable = 7.71
Hence, we reject H0 because 9.09 > 7.71
Selling Price ($) Suare Footage AGE Condition95000 1926 30 GOOD
119000 2069 40 Excellent124800 1720 30 Excellent135000 1396 15 GOOD142800 1706 32 Mint145000 1847 38 Mint159000 1950 27 Mint165000 2323 30 Excellent182000 2285 26 Mint183000 3752 35 GOOD200000 2300 18 GOOD211000 2525 17 GOOD215000 3800 40 Excellent219000 1740 12 Mint
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.819680305
R Square 0.671875802
Adjusted R Square 0.612216857
Standard Error 24312.60729
Observations 14
ANOVA
df SS MS F Significance F
Regression 2 13313936968 6.7E+09 11.262 0.002178765
Residual 11 6502131603 5.9E+08
Total 13 19816068571
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 146630.89 25482.08287 5.75427 0.0001 90545.20735 202717 90545 202717
SF 43.819366 10.28096507 4.26218 0.0013 21.19111495 66.448 21.191 66.448
AGE -2898.686 796.5649421 -3.639 0.0039 -4651.91386 -1145 -4651.9 -1145.5
The p-values are used to test the individual variables for significance
The coefficient of determination r2
The regression coefficients
Jenny Wilson Reality
Binary or Dummy Variables Indicator Variable Assigned a value of 1 if a particular condition is
met, 0 otherwise The number of dummy variables must equal one
less than the number of categories of a qualitative variable
The Jenny Wilson realty example :– X3= 1 for excellent condition
= 0 otherwise– X4= 1 for mint condition
= 0 otherwise
Selling Price ($) Suare Footage AGE X3(Exc.) X4(Mint) Condition
95000 1926 30 0 0 GOOD119000 2069 40 1 0 Excellent124800 1720 30 1 0 Excellent135000 1396 15 0 0 GOOD142800 1706 32 0 1 Mint145000 1847 38 0 1 Mint159000 1950 27 0 1 Mint165000 2323 30 1 0 Excellent182000 2285 26 0 1 Mint183000 3752 35 0 0 GOOD200000 2300 18 0 0 GOOD211000 2525 17 0 0 GOOD215000 3800 40 1 0 Excellent219000 1740 12 0 1 Mint
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.94762
R Square 0.89798
Adjusted R Square 0.85264
Standard Error 14987.6
Observations 14
ANOVA
df SS MS F Significance F
Regression 4 17794427451 4E+09 19.8044 0.000174421
Residual 9 2021641120 2E+08
Total 13 19816068571
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 121658 17426.61432 6.9812 6.5E-05 82236.71393 161080 82236.71 161080
SF 56.4276 6.947516792 8.122 2E-05 40.71122594 72.144 40.71123 72.144
AGE -3962.82 596.0278736 -6.6487 9.4E-05 -5311.12866 -2614.5 -5311.129 -2614.5
X3(Exc.) 33162.6 12179.62073 2.7228 0.0235 5610.432651 60714.9 5610.433 60715
X4(Mint) 47369.2 10649.26942 4.4481 0.0016 23278.92699 71459.6 23278.93 71460
The coefficients of age is negative, indicating that the price decreases as a house gets older
Jenny Wilson Reality
Model Building
The value of r2 can never decrease when more variables are added to the model
Adjusted r2 often used to determine if an additional independent variable is beneficial
The adjusted r2 is
A variable should not be added to the model if it causes the adjusted r2 to decrease
Multiple RegressionSales/Decision to buy = B0+ B1* Price
Sales/Decision to buy = B0+ B1* (Price)3+ B2*(Design)2+B3*(Performance)
L = (Price)3
M = (Design)2
N = (Performance)
Sales/Decision to buy = B0+ B1* L+ B2* M+ B3* N
Pitfalls In Regression
A High Correlation does not mean one variable is causing a change in another (Some regressions have shown a significantly positive relation between individuals' college GPA and future salary. )
Values of the dependent variable should not be used that are above or below the ones from the sample
The number of independent variables that should be used in the model is limited by the number of observations.
Thank
You!!!
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