Linear Regression Log Transformation

In mathematics, linear regression is one of the most important topics in statistics. The process of determining the relationship among the two variables is called as linear regression. It is also one of the statistical analysis methods that can be used to assessing the association between the two different variables. In this article, we shall discuss about the linear regression log transformation. It is used to help the students to know about the log transformation and the step by step explanations for the student doubts. The following are the example problem in linear regression log transformation.

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Linear regression log transformation - Formula:

Formula for linear regression:

Linear Regression Equation(y) = a + bln (x)
Slope (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
Intercept(a) = `(sumY - b(sumX)) / N`


Where
x and y are the variables.
b = the slope of the regression line is called as regression coefficient
a = intercept point of the regression line which is in the y-axis.
N = Number of values or elements
X = First Score
Y = Second Score
`sumXY` = Sum of the product of the first and Second Scores
`sumX` = Sum of First Scores
`sumY` = Sum of Second Scores
`sumX^2` = Sum of square First Scores.

Linear regression log transformation - Example problem:

Example:

Plot the linear regression for the given set of data and also find the regression equation by finding the regression coefficients and slope values.
x y
45 12
58 25
74 33
65 44
88 28
70 31
89 44
62 56
38 16
25 10


Solution:

Let us count the number of values.
N = 12

Determine the values for xy, x2, y2
x y xy x2 y2
45 12 540 2025 144
58 25 1450 3364 625
74 33 2442 5476 1089
65 44 2860 4225 1936
88 28 2464 7744 784
70 31 2170 4900 961
89 44 3916 7921 1936
62 56 3472 3844 3136
38 16 608 1444 256
25 10 250 625 100


Find the following values `sumX` , `sumY` , `sumXY` , `sumX^2` , `sumY^2` .
`sumX` = 614
`sumY` = 299
`sumXY` = 20172
`sumX^2 ` = 41568
`sumY^2` = 10967

Substitute values in the slope formula
Slope (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
= `((10)*(20172)-(614)*(299))/((10)*(41568)-(614)^2)`

= `(201720 -183586)/(415680-376996) `
=`18134/38684`
b = 0.46877
Substitute the values in the intercept formula given.
Intercept (a) = `(sumY - b (sumX)) / N`
= `(299- 0.46877 (614))/10`
= `(299 - 287.82478)/10`
= `11.1736/10`
a = 1.11736


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Substitute the Regression coefficient value and intercept value in the regression equation

Linear Regression Equation for log transformation (y) = a + bln(x)
= 1.11736 + 0.46877ln (x)

For the given set of data, we can plot the graph

1. Take the X and Y values in the given set of data

2. In the graph, take the scale as for x and y axis as follows

For x axis - 1 unit = 10cm

For y axis - 1 unit = 20 cm

3. Mark the x and y values in the graph

4. Plot the graph as in the diagram

Linear regression log transformation - Example