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How to Calculate Variance, Standard Error, and T-Value in Multiple Linear Regression

Finding variance, standard error, and t-value was an important stage to test the research hypothesis. The formula used in multiple linear regression is different from simple linear regression. On this occasion, I will discuss calculating the multiple linear regression with two independent variables.

The estimate of the variance of u must calculate first to obtain the variance. The sum of residual squared is one of the components that must be found first to calculate the value of the variance estimate.

As a material for the exercise, I used the same data as my previous article entitled: “How To Calculate bo, b1, and b2 Coefficient Manually in Multiple Linear Regression.”

Formula to Find Variance

To calculate variance, we must find the estimate of the variance of u first. To find the estimate of the variance of u, we can refer to the book written by Koutsoyiannis (1977):

Based on the estimate of the variance of u calculation formula, you must have calculated the sum of residual squared values. Based on our calculation results before in the article entitled: “How to Find Y Predicted, Residual, and Sum of Squares in Multiple Linear Regression“, the value of the residual squared sum was 124.36.

Furthermore, the number of observations (n) = 15 and the number of variables (K) = 3. After calculating using this formula, the estimate of the variance of u = 10.36 was obtained.

The variance in multiple linear regression was determined for bo, b1 and b2. The formula is based on the book written by Koutsoyiannis (1977), namely:

Based on the formula, the variance estimate of u was used to determine the variance value of bo, b1, and b2. The variance of bo, b1 and b2 values obtained 38.47179; 0.01333; and 0.00387, respectively.

Formula to Find Standard Error

The standard error can be determined after we calculate variance. The formula to find standard error refers to the book written by Koutsoyiannis (1977), which can be seen below:

Based on the formula above, the standard error is the square root of the variance. Based on the calculation results, the standard error of bo, b1, and b2 was 6.20256, 0.11545, and 0.06221, respectively.

Formula to Find T-Value

Finding the t-value needs the estimated coefficient and standard error. In detail, the formula to find the t-value refers to the book written by Koutsoyiannis (1977), namely:

Calculating the t-value for the intercept (bo), b1, and b2 was -3.05540, 3.79932, and 2.70440, respectively.

Coefficient Table Recap

Based on the results of the calculations that we have done, then I recap the coefficient table as follows:

You can conduct data processing using menu Data Analysis in Excel to correct the calculation results. I have conducted a multiple linear regression analysis using Excel. Excel output for a table containing standard error values and t-values can be seen in the image below:

Based on the Excel output, we can compare the results of our calculations. If our calculation result is exactly the same as the Excel output, there is no calculation error.

After I checked the calculation results above and then compared it with the Excel output, I concluded that the estimate of variance, standard error, and T-value were correct.

Well, that’s the tutorial that I can convey to all of you. Hopefully, it’s helpful to us. See you in the following article!

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