# bootci

Bootstrap confidence interval

## Syntax

## Description

creates each bootstrap sample by sampling with replacement from the rows of the nonscalar
data arguments in `ci`

= bootci(`nboot`

,`bootfun`

,`d`

1,...,`d`

N)`d1,...,dN`

. These nonscalar arguments must have the same
number of rows. The `bootci`

function passes the samples of nonscalar
data and the unchanged scalar data arguments in `d1,...,dN`

to
`bootfun`

.

specifies options using one or more name-value arguments. For example, you can change the
significance level of the confidence interval by specifying the `ci`

= bootci(`nboot`

,{`bootfun`

,`d`

1,...,`d`

N},`Name,Value`

)`'Alpha'`

name-value argument.

Note that you must pass the `bootfun`

and
`d1,...,dN`

arguments to `bootci`

as a single cell
array.

## Examples

### Bootstrap Confidence Interval

Compute the confidence interval for the capability index in statistical process control.

Generate 30 random numbers from the normal distribution with mean 1 and standard deviation 1.

rng('default') % For reproducibility y = normrnd(1,1,30,1);

Specify the lower and upper specification limits of the process. Define the capability index.

LSL = -3; USL = 3; capable = @(x)(USL-LSL)./(6*std(x));

Compute the 95% confidence interval for the capability index by using 2000 bootstrap samples. By default, `bootci`

uses the bias corrected and accelerated percentile method to construct the confidence interval.

ci = bootci(2000,capable,y)

`ci = `*2×1*
0.5937
0.9900

Compute the studentized confidence interval for the capability index.

sci = bootci(2000,{capable,y},'Type','student')

`sci = `*2×1*
0.5193
0.9930

### Bootstrap Confidence Intervals for Nonlinear Regression Model Coefficients

Compute bootstrap confidence intervals for the coefficients of a nonlinear regression model. The technique used in this example involves bootstrapping the predictor and response values, and assumes that the predictor variable is random. For a technique that assumes the predictor variable is fixed and bootstraps the residuals, see Bootstrap Confidence Intervals for Linear Regression Model Coefficients.

**Note:** This example uses `nlinfit`

, which is useful when you only need the coefficient estimates or residuals of a nonlinear regression model and you need to repeat fitting a model multiple times, as in the case of bootstrapping. If you need to investigate a fitted regression model further, create a nonlinear regression model object by using `fitnlm`

. You can create confidence intervals for the coefficients of the resulting model by using the `coefCI`

object function, although this function does not use bootstrapping.

Generate data from the nonlinear regression model $$y={b}_{1}+{b}_{2}\cdot exp(-{b}_{3}x)+\u03f5$$, where $${b}_{1}=1$$, $${b}_{2}=3$$, and $${b}_{3}=2$$ are coefficients; the predictor variable *x* is exponentially distributed with mean 2; and the error term $$\u03f5$$ is normally distributed with mean 0 and standard deviation 0.1.

modelfun = @(b,x)(b(1)+b(2)*exp(-b(3)*x)); rng('default') % For reproducibility b = [1;3;2]; x = exprnd(2,100,1); y = modelfun(b,x) + normrnd(0,0.1,100,1);

Create a function handle for the nonlinear regression model that uses the initial values in `beta0`

.

beta0 = [2;2;2]; beta = @(predictor,response)nlinfit(predictor,response,modelfun,beta0)

`beta = `*function_handle with value:*
@(predictor,response)nlinfit(predictor,response,modelfun,beta0)

Compute the 95% bootstrap confidence intervals for the coefficients of the nonlinear regression model. Create the bootstrap samples from the generated data `x`

and `y`

.

ci = bootci(1000,beta,x,y)

`ci = `*2×3*
0.9821 2.9552 2.0180
1.0410 3.1623 2.2695

The first two confidence intervals include the true coefficient values $${b}_{1}=1$$ and $${b}_{2}=3$$, respectively. However, the third confidence interval does not include the true coefficient value $${b}_{3}=2$$.

Now compute the 99% bootstrap confidence intervals for the model coefficients.

`newci = bootci(1000,{beta,x,y},'Alpha',0.01)`

`newci = `*2×3*
0.9730 2.9112 1.9562
1.0469 3.1876 2.3133

All three confidence intervals include the true coefficient values.

### Bootstrap Confidence Intervals for Linear Regression Model Coefficients

Compute bootstrap confidence intervals for the coefficients of a linear regression model. The technique used in this example involves bootstrapping the residuals and assumes that the predictor variable is fixed. For a technique that assumes the predictor variable is random and bootstraps the predictor and response values, see Bootstrap Confidence Intervals for Nonlinear Regression Model Coefficients.

**Note:** This example uses `regress`

, which is useful when you only need the coefficient estimates or residuals of a regression model and you need to repeat fitting a model multiple times, as in the case of bootstrapping. If you need to investigate a fitted regression model further, create a linear regression model object by using `fitlm`

. You can create confidence intervals for the coefficients of the resulting model by using the `coefCI`

object function, although this function does not use bootstrapping.

Load the sample data.

`load hald`

Perform a linear regression and compute the residuals.

x = [ones(size(heat)),ingredients]; y = heat; b = regress(y,x); yfit = x*b; resid = y - yfit;

Compute the 95% bootstrap confidence intervals for the coefficients of the linear regression model. Create the bootstrap samples from the residuals. Use normal approximated intervals with bootstrapped bias and standard error by specifying `'Type','normal'`

. You cannot use the default confidence interval type in this case.

ci = bootci(1000,{@(bootr)regress(yfit+bootr,x),resid}, ... 'Type','normal')

`ci = `*2×5*
-47.7130 0.3916 -0.6298 -1.0697 -1.2604
172.4899 2.7202 1.6495 1.2778 0.9704

Plot the estimated coefficients `b`

, omitting the intercept term, and display error bars showing the coefficient confidence intervals.

```
slopes = b(2:end)';
lowerBarLengths = slopes-ci(1,2:end);
upperBarLengths = ci(2,2:end)-slopes;
errorbar(1:4,slopes,lowerBarLengths,upperBarLengths)
xlim([0 5])
title('Coefficient Confidence Intervals')
```

Only the first nonintercept coefficient is significantly different from 0.

### Confidence Intervals for Multiple Statistics

Compute the mean and standard deviation of 100 bootstrap samples. Find the 95% confidence interval for each statistic.

Generate 100 random numbers from the exponential distribution with mean 5.

rng('default') % For reproducibility y = exprnd(5,100,1);

Draw 100 bootstrap samples from the vector `y`

. For each bootstrap sample, compute the mean and standard deviation. Find the 95% bootstrap confidence interval for the mean and standard deviation.

[ci,bootstat] = bootci(100,@(x)[mean(x) std(x)],y);

`ci(:,1)`

contains the lower and upper bounds of the mean confidence interval, and `c(:,2)`

contains the lower and upper bounds of the standard deviation confidence interval. Each row of `bootstat`

contains the mean and standard deviation of a bootstrap sample.

Plot the mean and standard deviation of each bootstrap sample as a point. Plot the lower and upper bounds of the mean confidence interval as dotted vertical lines, and plot the lower and upper bounds of the standard deviation confidence interval as dotted horizontal lines.

plot(bootstat(:,1),bootstat(:,2),'o') xline(ci(1,1),':') xline(ci(2,1),':') yline(ci(1,2),':') yline(ci(2,2),':') xlabel('Mean') ylabel('Standard Deviation')

## Input Arguments

`nboot`

— Number of bootstrap samples

positive integer scalar

Number of bootstrap samples to draw, specified as a positive integer scalar. To
create each bootstrap sample, `bootci`

randomly selects with
replacement `n`

out of the `n`

rows of nonscalar data
in `d`

or `d1,...,dN`

.

**Example: **`100`

**Data Types: **`single`

| `double`

`bootfun`

— Function to apply to each sample

function handle

Function to apply to each sample, specified as a function handle. The function can
be a custom or built-in function. You must specify `bootfun`

with the
`@`

symbol.

**Example: **`@mean`

**Data Types: **`function_handle`

`d`

— Data to sample from

column vector | matrix

Data to sample from, specified as a column vector or matrix. The
`n`

rows of `d`

correspond to observations. When
you use multiple data input arguments `d1,...,dN`

, you can specify some
arguments as scalar values, but all nonscalar arguments must have the same number of
rows.

If you use a single vector argument `d`

, you can specify it as a
row vector. `bootci`

then samples from the elements of the
vector.

**Data Types: **`single`

| `double`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`bootci(100,{@mean,1:6'},'Alpha',0.1)`

specifies to draw 100
bootstrap samples from the values 1 through 6, take the mean of each sample, and then
compute the 90% confidence interval for the sample mean.

`Alpha`

— Significance level

`0.05`

(default) | positive scalar in (0,1)

Significance level, specified as a positive scalar between 0 and 1.
`bootci`

computes the `100*(1-Alpha)`

bootstrap
confidence interval of each statistic defined by the function
`bootfun`

.

**Example: **`'Alpha',0.01`

**Data Types: **`single`

| `double`

`Type`

— Confidence interval type

`'bca'`

(default) | `'norm'`

| `'per'`

| `'cper'`

| `'stud'`

| ...

Confidence interval type, specified as one of the values in this table.

Value | Description |
---|---|

`'norm'` or `'normal'` | Normal approximated interval with bootstrapped bias and standard error [1] |

`'per'` or `'percentile'` | Basic percentile method |

`'cper'` or ```
'corrected
percentile'
``` | Bias corrected percentile method [2] |

`'bca'` | Bias corrected and accelerated percentile method [3], [4]. This method involves a
z by
including half of the bootstrap values that are the same as the original
sample value._{0} |

`'stud'` or `'student'` | Studentized confidence interval [3] |

**Example: **`'Type','student'`

`NBootStd`

— Number of bootstrap samples for studentized standard error estimate

`100`

(default) | positive integer scalar

Number of bootstrap samples for the studentized standard error estimate, specified as a positive integer scalar.

`bootci`

computes the studentized bootstrap confidence interval
of the statistic defined by the function `bootfun`

, and estimates
the standard error of the bootstrap statistics by using `NBootStd`

bootstrap data samples.

**Note**

To use this name-value argument, the `Type`

value must be
`'stud'`

or `'student'`

. Specify either
`NBootStd`

or `StdErr`

, but not both.

**Example: **`'NBootStd',50`

**Data Types: **`single`

| `double`

`StdErr`

— Function used to compute studentized standard error estimate

function handle

Function used to compute the studentized standard error estimate, specified as a function handle.

`bootci`

computes the studentized bootstrap confidence interval
of the statistic defined by the function `bootfun`

, and estimates
the standard error of the bootstrap statistics by using the function
`StdErr`

. The `StdErr`

function must take the
same arguments as `bootfun`

and return the standard error of the
statistic computed by `bootfun`

.

**Note**

To use this name-value argument, the `Type`

value must be
`'stud'`

or `'student'`

. Specify either
`NBootStd`

or `StdErr`

, but not both.

**Example: **`'StdErr',@std`

**Data Types: **`function_handle`

`Weights`

— Observation weights

`ones(n,1)/n`

(default) | nonnegative vector

Observation weights, specified as a nonnegative vector with at least one positive
element. The number of elements in `Weights`

must be equal to the
number of rows `n`

in the data `d`

or
`d1,...,dN`

. To obtain one bootstrap sample,
`bootci`

randomly selects with replacement `n`

out of `n`

rows of data using these weights as multinomial sampling
probabilities.

**Data Types: **`single`

| `double`

`Options`

— Options for computing in parallel and setting random streams

structure

Options for computing in parallel and setting random streams, specified as a
structure. Create the `Options`

structure using `statset`

. This table lists the option fields and their
values.

Field Name | Value | Default |
---|---|---|

`UseParallel` | Set this value to `true` to run computations in
parallel. | `false` |

`UseSubstreams` | Set this value to To compute
reproducibly, set | `false` |

`Streams` | Specify this value as a `RandStream` object or
cell array of such objects. Use a single object except when the
`UseParallel` value is `true`
and the `UseSubstreams` value is
`false` . In that case, use a cell array that
has the same size as the parallel pool. | If you do not specify `Streams` , then
`bootci` uses the default stream or
streams. |

**Note**

You need Parallel Computing Toolbox™ to run computations in parallel.

**Example: **`Options=statset(UseParallel=true,UseSubstreams=true,Streams=RandStream("mlfg6331_64"))`

**Data Types: **`struct`

## Output Arguments

`ci`

— Confidence interval bounds

vector with two rows | matrix with two rows | multidimensional array with two rows

Confidence interval bounds, returned as a vector, matrix, or multidimensional array with two rows.

If

`bootfun`

returns a scalar, then`ci`

is a vector containing the lower and upper bounds of the confidence interval.If

`bootfun`

returns a vector of length*m*, then`ci`

is a matrix of size 2-by-*m*, where`ci(1,:)`

are lower bounds and`ci(2,:)`

are upper bounds.If

`bootfun`

returns a multidimensional array, then`ci`

is an array, where`ci(1,:,...)`

is an array of lower bounds and`ci(2,:,...)`

is an array of upper bounds.

`bootstat`

— Bootstrap statistics

column vector | matrix

Bootstrap statistics, returned as a column vector or matrix with
`nboot`

rows. The `i`

th row of
`bootstat`

corresponds to the results of applying
`bootfun`

to the `i`

th bootstrap sample. If
`bootfun`

returns a matrix or array, then the
`bootci`

function first converts this output to a row vector before
storing it in `bootstat`

.

## References

[1] Davison, A. C., and D. V. Hinkley. *Bootstrap Methods and Their Applications.* Cambridge University Press, 1997.

[2] Efron, Bradley. *The Jackknife, the Bootstrap and Other Resampling Plans.* Philadelphia: The Society for Industrial and Applied Mathematics, 1982.

[3] DiCiccio, Thomas J., and Bradley Efron. “Bootstrap Confidence Intervals.” *Statistical Science* 11, no. 3 (1996): 189–228.

[4] Efron, Bradley, and Robert J. Tibshirani. *An Introduction to the Bootstrap*. New York: Chapman & Hall, 1993.

## Extended Capabilities

### Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, specify the `Options`

name-value argument in the call to
this function and set the `UseParallel`

field of the
options structure to `true`

using
`statset`

:

`Options=statset(UseParallel=true)`

For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

## Version History

**Introduced in R2006a**

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