Тест бокса кокса в excel

We seek a transformation of data in a sample x₁, …, x_n which results in data that is normally distributed.

If one of the sample values is not positive, then we add 1–a to all the sample values where a is the smallest sample value. This will transform the sample into only positive values. Next, we sort the sample into ascending order, and so assume that x₁ ≤ … ≤ x_n. We now define the Box-Cox transformation y₁, …, y_n for any value of λ ≠ 0

We could also define the transformation at λ = 0, but because

this is not necessary. In fact, if in the optimization process described below, we get a value of λ close to zero, we will generally use the value λ = 0 to get a ln transformation.

We now use the approach described in Graphical Tests for Normality to create QQ plots by defining a sequence z₁, …, z_n where

and Φ^-1 is the inverse of the standard normal distribution, i.e. Φ^-1(p) = NORM.S.INV(p).

Finally, we use Excel’s Goal Seek to find the value of λ which maximizes the correlation coefficient between the y and z values.

Example 1: Find the Box-Cox transformation which best normalizes the data in range B4:B13 of Figure 1.

Figure 1 – Non-normal data

As we can see from the QQ plot and the results of the Shapiro-Wilk test, this data is not normal. We now create the x, y and z values for the data, as described above. This is shown in Figure 2.

Figure 2 – Set up for Goal Seek

Here, column O consists of the data in sorted order, e.g. by using the formula =QSORT(N4:N13), and column P contains the transformed data, e.g. cell P4 contains the formula =(O4^V$4-1)/V$4 or optionally =IF(V4=0,LN(O4),(O4^V$4-1)/V$4). Column S consists of the inverse normal values, and so cell S4 contains the formula =NORM.S.INV((R4-0.5)/R$13). Finally, cell V4 contains a guess for the λ value and cell V5 contains the formula =CORREL(P4:P13,S4:S13) to calculate the correlation coefficient between the y and z values.

We now select Data > Data Tools|What-If Analysis and choose the Goal Seek tool. Next, we fill in the dialog box that appears with the values shown on the right side of Figure 2 and press the OK button. The result is that the value for λ converges to -0.286629.

Figure 3 – Goal Seek output

Figure 3 contains the results of the Shapiro-Wilk test and QQ plot on the transformed data in column P of Figure 2. As you can see, the transformed data is normally distributed.

Figure 4 – Testing normality for transformed data

Actually the value λ = -0.216 yields a slightly better correlation, namely r = .9937987, but the normality testing is not much different for this transformation.

Another approach to optimizing the value of lambda is to maximize the log-likelihood function, which in this case is

where s² = the population variance of the transformed xi values, as calculated using the VAR.P(R1) function in Excel. The value of LL can be maximized using Goal Seek or using the golden search algorithm. The Real Statistics Resource Pack has implemented this approach as described in Example 2.

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions:

BOXCOX(R1, λ): array function which returns a range containing the Box-Cox transformation of the data in range R1 using the given lambda value. If the lambda argument is omitted, then the transformation which best normalizes the data in R1 is used, based on maximizing the log-likelihood function.

BOXCOXLL(R1, λ) = log likelihood function of the Box-Cox transformation of the data in R1 using the given lambda value

BOXCOXLambda(R1) = the value of lambda which maximizes the log likelihood function of the Box-Cox transformation of the data in R1

Example 2: Repeat Example 1 using the Real Statistics functions

We begin by displaying the Box-Cox transformation for values of lambda between -2 and 2, as shown in Figure 5. E.g. the transform of the data element when λ = -2 are shown in the range D4:D13, as calculated by the array formula =BOXCOX($B$4:$B$13,D3).

The LL value when λ = -2 is -398802 (cell D14), as calculated by the formula =BOXCOXLL($B$4:$B$13,D3). Alternatively, we could use the formula

=- COUNT(B4:B13)/2*LN(VARP(D4:D13))+(D3-1)*SUMPRODUCT(LN(B4:B13))

We see from Figure 5 that the maximum value of LL occurs somewhere between λ = -.5 and λ = 0. We can make successive guesses to eventually find a value sufficiently close to the value of λ that maximizes LL sufficiently well for our needs. Essentially this is what the BOXCOXLambda function does.

Figure 5 – Box-Cox optimization

In fact, we see that =BOXCOXLambda(B4:B13) = -0.16394 (cell N3) provides this value of lambda, corresponding to LL = -24.8913 (cell N14). The transformation values are shown in range N4:N13, as calculated by the array function =BOXCOX(B4:B13), which is equivalent to the array function =BOXCOX(B4:B13, N3).

Finally, we note that the optimal value λ = -0.286629 that we found in Example 1 yields an LL value which is almost as good (LL = -24.9853), but the LL value for λ  = -0.16394 is slightly better.