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ISYE 6414 - Midterm Exam WITH 100- CORRECT ANSWER
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ISYE 6414 - Midterm Exam WITH 100- CORRECT ANSWER
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10/6/24, 8:03 AMEUNICE
ISYE 6414 - Midterm Exam WITH QUESTIONS AND 100%
SURE ANSWERS
Regression analysis is a simple way to investigate the relationship between 2 or more
Regression Analysis
variables in a non-deterministic way.
This is a variable we're interested in understanding, modeling or testing
Response/Target Variable (Y)
This is a random variable. It varies with changes in the predictor(s)
These are variables we think might be useful in predicting or modeling the response
2. Predicting/Explanatory (independent) variable
Variables(Xs ~ X1, X2)
This is a fixed variable. It does not change with the response
We have a straight line which doesn't fit perfectly to the points
The objective is to fit a non-deterministic linear model between the predicting
Simple Linear Regression
variable and Y.
In simple linear regression, we have 3 parameters to estimate.
Multiple Linear Regression We can have a plane if we have two predictions
Polynomial Regression We are capturing a nonlinear relationship
1. Prediction: We want to see how the response variable behaves in different settings
2. Modeling: We are interested in modeling the relationship between the response
Objectives of Linear Regression variable and the explanatory/predicting variables
3. Testing: We are also interested in testing the hypotheses of association
relationships.
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• Linearity/Mean Zero Assumption: This means that the expected value of the errors
is zero
• Constant Variance Assumption: This means that the variance of the error term is
equal to sigma_squared is the same across all error terms
Simple Linear Regression Assumptions • Independence Assumption: This means that the error terms are independent
random variables i.e. deviances (response variables Ys) are independently drawn
from the data generating process -- it cannot be true that the model under-predicts
Y for one particular case tells you anything or all about what it does for any other
case
• Normal Assumption: The errors are assumed to be normally distributed.
A violation of this assumption will lead to difficulties in estimating 0 and means that
Linearity Assumption
your model does not include a necessary systematic component
This means that the model cannot be more accurate in some parts and less accurate
in other parts of the model. The variance has to be constant.
Constant Variance Assumption
A violation of this assumption means that the estimates are not as efficient as they
could be in estimating the true parameters and better estimates can be calculated
also results in poorly calibrated prediction intervals
It cannot be true that the model under-predicts Y. One particular case doesn't tell
you anything or all about what it does for any other case
This violation most often occurs in data that are ordered in time (time series data)
Independence Assumption
where areas that are near each other in time are similar to each other.
Violation of this assumption can lead to very misleading assessments of the strength
of the regression
This is needed if we want to do any confidence or prediction intervals, or hypothesis
test
Normal Assumption
If this assumption is violated, hypothesis test and confidence and prediction intervals
can be very misleading. This assumption is needed got statistical inference.
Autocorrelation Time-related correlation is often called autocorrelation.
The error term is also a parameter in linear regression. Epsilon is the deviance of the
data from the linear model.
Error term
The error term is also normally distributed
Sigma_squared is a chi-squared distribution with n-2 degrees of freedom
Sample Variance Estimation
For the sample variance, we lose one degree of freedom from sigma_squared. We
end up with a chi-squared distribution with n-1 degrees of freedom
a positive value for Beta_1, then that's consistent with a direct relationship between
the predicting variable X and the response variable Y
Model Parameter Interpretation negative value of Beta_1 is consistent with an inverse relationship between X and Y
When Beta_1 is close to zero, we interpret that there is not a significant association
between the predicting variable X and the response variable Y.
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