Statistical Analysis Report: Simple and Multiple Linear Regression

This report delves into the concepts of simple and multiple linear regression, providing a clear understanding of their applications and underlying principles. The report begins by explaining simple linear regression, using a case study of a hotel service to illustrate how to predict a dependent variable (tip amount) based on an independent variable (bill amount). It highlights key concepts like residuals, SSE (Sum of the Squared Errors), and the objective of minimizing SSE to achieve the best-fit line. The report then transitions to multiple linear regression, where multiple independent variables are used to predict the dependent variable. It addresses potential issues such as overfitting and multicollinearity, explaining how to build an efficient model by selecting relevant independent variables and conducting multicollinearity tests. Finally, the report describes the interpretation of coefficients in multiple linear regression, emphasizing the predicted change in the response variable for a one-unit change in an independent variable while holding other variables constant.

Statistics
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Statistics 101: Simple Linear Regression, The Very Basics
Link: https://www.youtube.com/watch?v=ZkjP5RJLQF4
Summary
To clarify the concept behind simple linear regression a case study of hotel service is
used. In this article, the hotel owner need to estimate the tip expected from any given bill
amount. To do the estimate the tip amount values which are the dependent variable are collected.
Using only the dependent variable, the hotel owner have to come up with a model that best
predicts the next expected tip amount. Using this single variable with no other additional
information, the best estimate for the tip at any given bill amount will be given by the mean. The
dependent variable which in this case is the tip amount is the variable that is to be predicted.
The difference between the value predicted using the model and the actual recorded
value is known as the residual. The sum of all the residual values do add up to zero. The
residuals are squared to emphasize on the larger deviations. The sum of the squared residuals is
what is known as SSE (Sum of the Squared Errors). The objective of the simple linear regression
is to develop a linear model that minimizes the SSE. This is what is defined as the best fit line.
When the independent variable is introduced, a significant regression model is expected to
reduce the SSE that was obtained when only the dependent variable data was used. The
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independent variable will minimize the residuals. So, the fit of a model is a comparison of the
best fit line that is obtained when the independent variable is introduced to the best fit line that
was obtained using the dependent variable only.
Statistics 101: Multiple Linear Regression, The Very Basics
Link: https://www.youtube.com/watch?v=dQNpSa-bq4M
Summary
Multiple regression is an extension of the simple linear regression only that in the case of
the multiple linear regression there are two or more independent variables that are used in
predicting the dependent variable. The use of many independent variables may give rise to two
forms of problems one is overfitting and the other is multicollinearity. Overfitting occurs when
too many independent variables are introduced to the model. This accounts for more variance but
adds little value to the goodness of the model. To come up with an efficient model, only the
independent variables that add value should be added to the model this way the overfitting issue
is limited. On the other hand, multicollinearity happens when some or all of the independent
variables are correlated with each other. Due to the correlation among the independent variable it
may not be possible to estimate which of the independent variables do correlate to the dependent
variable. So, prior to fitting a multiple regression model, a multicollinearity test has to be
conducted to eliminate the independent variables that are highly correlated. In the case of the
multiple linear repression each of the coefficients is interpreted as the predicted change in the
response variable that corresponds to a one unit change in the independent variable taking all the
other variables of the model as constants.
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