Jackknife

Overview

Assume that we have a random sample S = {x₁, …, x_n} and^θ is the estimate of some parameter θ based on this sample using some function f(S).

Now define S_(i) to be S with x_i removed and^θ_(i) = f(S_(i)) to be the estimate of θ based on the sample S_(i). The jackknife sample consists of the set^θ₍₁₎, …,^θ_(n). We also define the jackknife estimate^θ₍₀₎ of θ to be the mean of the jackknife sample

The jackknife can be used to estimate the standard error of the estimate as the population standard deviation of the jackknife sample multiplied by the square root of n–1

Acceleration

The acceleration coefficient is defined based on a jackknife sample by

We will use the acceleration to create the biased-corrected and accelerated (BCa) bootstrap confidence interval. See Bootstrapping for more information about acceleration.

Worksheet Function

Real Statistics Function: The Real Statistics Resource Pack provides the following lambda array function.

JACKKNIFE(R1, expression, ref): returns a column array with the jackknife sample for the data in R1 based on the function f(arr) on a variable arr that takes array values; expression is used to specify f and ref is an optional reference to arr.

Acceleration(R1, expression, ref) = the acceleration for the sample data in R1 based on the function f(arr) on a variable arr that takes array values; expression is used to specify f and ref is an optional reference to arr.

If R1 is a column array then its elements x_i are those in the sample S. If R1 is an array with multiple columns then the same approach is used except that now the elements x_i represent rows in R1.

See Real Statistics Lambda Capabilities for additional information about lambda functions.

Example

Example 1: Use the jackknife to estimate the standard error of the mean based on the data in column B of Figure 1.

Figure 1 – Jackknife example

The parameter θ is the population mean and the sample S is the set of elements in range B2:B13. The elements in column C consist of the means of the S_(i). This can be done by placing the array formula =AVERAGE(DELROW(B$2:B$13,A2)) in cell C2, highlighting range C2:C13, and pressing Ctrl-D. DELROW is a Real Statistics function. If you want to use only Excel functions, you can place the formula =AVERAGE(B3:B13) in cell C2 and =AVERAGE(B2:B12) in C13. Next, you place the array formula

=AVERAGE(INDEX(B$2:B$13,A$2):INDEX(B$2:B$13,A2),INDEX(B$2:B$13,A4):INDEX(B$2:B$13,A$13))

in cell C3, highlight range B3:B12 and press Ctrl-D. Of course, you can also use the JACKKNIFE function, by placing the array formula

=JACKKNIFE(B2:B13,”=AVERAGE($arr)”)

in range B2:B13. Since JACKKNIFE is a lambda function, there are many alternative formations, such as =JACKKNIFE(B2:B13,”=AVERAGE(arr)”,”arr”).

Note that the jackknife mean (cell F4) is equal to the sample mean (cell F3). This won’t always be the case, though (e.g. if f(S) = the variance of S). The standard error of the mean is shown in cell F5. We can also obtain the standard error of the mean by using the JACKKNIFE function, as shown in cell F9. We will explain the acceleration value shown in cells F6 and F10 shortly.

Since the data in column B is approximately normally distributed (p-value = .3387 for the Shapiro-Wilk test), we expect that the standard error of 2.548024 (cell F5) will be quite close to the usual normal approximation of 2.548024 as calculated by =STDEV.S(B2:B13)/SQRT(COUNT(B2:B13)). In fact, for θ = the population mean, they are exactly the same. Again, this won’t always be the case for other population parameters.

Example using a multicolumn array

Example 2: Use jackknifing to estimate the standard error of the population correlation coefficient based on the sample of 8 subjects with measures before and after an intervention, as shown in range B2:C9 of Figure 2.

Figure 2 – Jackknife standard error for correlation

The correlation coefficient for the sample data is .882425 (cell H2), indicating a high degree of correlation. We estimate the standard error for the correlation using the jackknife sample shown in column E. We can obtain these values by inserting the array formula

=CORREL(INDEX(DELROW(B$2:C$9,A3),,1),INDEX(DELROW(B$2:C$9,A3),,2))

in cell E2, highlighting range E2:E9, and pressing Ctrl-D. alternatively, we can insert the lambda formula

=JACKKNIFE(B2:C9,”=CORREL(INDEX($arr,,1),INDEX($arr,,2))”)

in cell E2. Note that the delete-one-element approach is applied one row at a time to the array B2:C9.

We see that the standard error is estimated to be 0.136476 (cell H5 or H8). Note that this time the sample correlation and jackknife correlations (H2 and H3) are different.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

DiCiccio, T. J., Efron, B. (1996) Bootstrap confidence intervals.
http://staff.ustc.edu.cn/~zwp/teach/Stat-Comp/Efron_Bootstrap_CIs.pdf

Efron, B., Tibshirani, R. J., (1993) An introduction to the bootstrap. Springer
https://books.google.it/books/about/An_Introduction_to_the_Bootstrap.html?id=gLlpIUxRntoC&redir_esc=y