Performing Polynomial Regression in Excel

Polynomial regression is a type of regression analysis where the relationship between the independent variable (X) and dependent variable (Y) can be modeled as an nth degree polynomial function. In Excel, you can perform polynomial regression using the LINEST function.

What is the LINEST Function?

The LINEST function in Excel is a versatile tool for linear and polynomial regression analysis. It allows you to compute the coefficients of a regression equation, including polynomial regression. You can use it when you need more than just the polynomial equation; it provides additional statistical information like R-squared values, standard errors, and more.

Organize Your Data

As with any regression analysis, start by organizing your data into two columns: one for the independent variable (X) and another for the dependent variable (Y). Place your data in adjacent columns, typically in columns A and B.

Using the LINEST Function for Polynomial Regression

  1. Enter the Polynomial Powers: In an empty column (let’s say, column C), calculate the values of X raised to the power of the polynomial order. For example, if you’re conducting a quadratic regression, calculate X^2 for each data point. You can use a formula like =A2^2 in cell C2 and drag it down to apply to all data points.
  2. Select a Range for Results: Choose a range where you want to display the results of your polynomial regression analysis. This could be on the same worksheet or a different one. Let’s say you select cells E1:H4 for this example.
  3. Use the LINEST Function: Enter the following formula in cell E1:=LINEST(B2:B6, C2:C6^{1,2,3}, TRUE, TRUE)
    • B2:B6 represents the range of cells containing the dependent variable (Y) values.
    • C2:C6^{1,2,3} represents an array formula where the polynomial powers (1, 2, 3) correspond to the order of the polynomial. You can adjust this array to match the order of your polynomial.
    • TRUE as the third argument includes a constant term in the regression equation.
    • TRUE as the fourth argument returns additional statistical information.
  4. Enter as an Array Formula: Instead of pressing Enter, you should enter this formula as an array formula by pressing Ctrl+Shift+Enter. If entered correctly, Excel will surround the formula with curly braces {}.
  5. Interpret the Results: The array formula you entered in cell E1 will populate cells E1:H4 with various pieces of information, including the coefficients of the polynomial equation, the standard errors, the R-squared value, and more.
  6. Extract the Coefficients: You can access specific coefficients from the LINEST result array. For instance, the coefficients for your polynomial equation can be found in cells E1:H1. These coefficients represent the terms in your polynomial equation, starting from the constant term and ascending powers of X.
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By following these steps, you can utilize the LINEST function in Excel to perform polynomial regression and obtain detailed statistical information about the fit.

This approach is particularly useful when you require a deeper understanding of the relationship between your variables and need to evaluate the significance of coefficients and goodness-of-fit statistics.