Here we consider the joint estimation of a multivariate set of population means. That is, we have observed a set of p X-variables and may wish to estimate the population mean for each variable. In some instances, we may also want to estimate one or more linear combinations of population means. Our basic tool for estimating the unknown value of a population parameter is a confidence interval, an interval of values that is likely to include the unknown value of the parameter.
- General Format for a Confidence Interval
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The general format of a confidence interval estimate of a population mean is:
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\(\text{Sample mean} \pm \text{Multiplier × Standard error of mean}\)
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For variable \(X_{j}\), a confidence interval estimate of its population mean \(\mu_{j}\) is
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\(\bar{x}_j \pm \text{Multiplier}\dfrac{s_j}{\sqrt{n}}\)
In this formula, \(\bar{x}_{j}\) is the sample mean, \(s_{j}\) is the sample standard deviation and n is the sample size. The multiplier value is a function of the confidence level, the sample size, and the strategy used for dealing with the multiple inference issue.
Strategies for Determining the Multiplier Section
The following list covers some common strategies:
- One-at-a-Time Confidence Intervals: This strategy essentially considers each mean separately and uses the desired confidence level (usually 95%) for every single interval.
- Bonferroni Method: With this method, we set a family-wide error rate and then divide this family error rate by the number of intervals to be computed to determine the error rate (and hence confidence level) for each individual interval.
- Simultaneous Confidence Region: This strategy uses properties of the multivariate normal distribution to define joint confidence intervals. The multiplier for this method is conservative because the family error rate applies to the family of all possible linear combinations of population means.
One at a Time Intervals Section
For a \(1 - \alpha\) confidence interval, the “one at a time” multiplier is the t-value such that the probability is \(1 - \alpha\) between –t and +t under a t-distribution with n - 1 degrees of freedom. Said another way, the value of t is such that the probability greater than +t is \(\alpha/2\).
Notationally, the one-at-a-time multiplier is:
\(\text{Multiplier} = t_{n-1}(\alpha/2)\)
With this notation, a confidence interval for \(\mu_{j}\) is computed as:
\(\bar{x}_j \pm t_{n-1}(\alpha/2)\frac{s_j}{\sqrt{n}}\)
Example 5-1: One at a Time Intervals
Suppose that the sample size is n = 25 and we want a 95% confidence interval for the population mean. Thus \(\alpha = 0.05\). Our textbook would write the multiplier as \(t_{24}(.025)\). In Excel, the command =TINV(.05,24) will give the multiplier (value = 2.064). In SAS, a command such as t1=tinv(.975,24) will make the variable t1 that contains the desired multiplier.
Bonferroni Method Multiplier Section
When we determine confidence intervals for the population means of several variables, we are creating a family of confidence intervals. The family-wide error rate is the probability that at least one of the confidence intervals in the family will not capture the population mean. The family-wide confidence level = 1 – family-wide error rate.
Suppose that we have a family of p confidence intervals and the error rates for the individual intervals are \(\alpha _ { 1 }, \alpha _ { 2 }, \dots , \alpha _ { p }\). The Bonferroni Inequality states that the family wide-error rate is less than or equal to the sum of \(\alpha _ { 1 }, \alpha _ { 2 }, \dots , \alpha _ { p }\). That is family-wide error rate \(\leq \Sigma \alpha _ { i }\). In terms of the family-wide confidence that all intervals capture their population means, we can write this as \(1 - \Sigma \alpha _ { i } \leq\) family-wide confidence level.
Most often, we divide the desired family-wide error rate equally across the intervals that we will compute. If we are computing p confidence intervals with a desired family-wide confidence level of \(\alpha\), we use an error rate of \(\alpha / p\) (so confidence \(= 1 - (\alpha / p)\) for each individual interval. This guarantees that the family-wide confidence level will be greater than or equal to \(1 - \alpha\).
Suppose that we are calculating p intervals with a family error rate equal to \(\alpha\).
Notationally, the Bonferroni method multiplier is:
\(\text{Multiplier} = t_{n-1}(\alpha/2p)\)
A confidence interval for\(\mu_{j}\) is computed as:
\(\bar{x}_j \pm t_{n-1}(\alpha/2p)\frac{s_j}{\sqrt{n}}\)
Example 5-2: Bonferroni Method Multiplier
Suppose that n = 25. The family-wide error = 5% for a family confidence = 95%. We are computing intervals for p = 5 means. The error rate for each interval will be .05/5 = 1%. We might use the Excel command = TINV(.01,24) to find that the multiplier = 2.797. In SAS, we use the cumulative probability \(= 1- \alpha /2p\) so the command for finding the t-multiplier in this instance is something like t1=tinv(.995, 24).
Simultaneous Confidence Region Multiplier Section
This method is derived from the properties of the multivariate normal distribution. The multiplier applies to the family of all possible linear combinations of the population means considered, including the individual means. It is conservative (meaning that the multiplier tends to be larger than absolutely necessary). When family confidence is used, compare the value of this multiplier to the Bonferroni method multiplier and use the smaller of the two.
Notationally, the simultaneous confidence region multiplier is:
\(\text{Multiplier}=\sqrt{\frac{p(n-1)}{n-p}F_{p,n-p}(\alpha)}\)
\(F _ { p , n - p } ( \alpha )\) represents a value of F such that the probability greater than this value is α under an F-distribution with p and n - p degrees of freedom.
Example 5-3: Simultaneous Confidence Region Multiplier
Suppose that we have a sample size of n = 25 and we have p = 3 variables. With a 5% family error rate (and 95% family confidence), the F-value can be found in Excel using = FINV(.05, 3, 22) = 3.049. SAS uses cumulative probabilities so in this case, a command like f1= FINV(.95,3, 22) would make f1 the F-value. The multiplier in this example is
\(\sqrt{\frac{3(25-1)}{25-3}3.049}=3.159\)
This multiplier could be used for all confidence intervals for parameters that are linear combinations of the three population means (and for the three individual means).
Summary of Multipliers Section
The following table summarizes the three different multipliers and gives notes about using Excel and SAS.
| Method | Textbook notation for multiplier | Excel notes | SAS notes |
|---|---|---|---|
| One at a time: Confidence = (\(1 - \alpha)\) for each interval | \(t _ { n - 1 } ( \alpha / 2 )\) | To determine the t -value, enter the equation TINV(\(\alpha, \text{df}\)) |
To determine the t-value, create t1= tinv(\(1 - \alpha/2,\ n-1\)) |
|
Bonferroni Method: Confidence = \(1 - \alpha)\) for whole family |
\(t _ { n - 1 } ( \alpha / 2 p )\) |
To determine the t -value, enter the equation TINV(α / p, df) |
To determine the t- value, create t2= tinv(\(1 - \alpha / 2p,\ n-1\)) |
| Multivariate Simultaneous Intervals |
\(\sqrt{\frac{p(n-1)}{n-p}F_{p,n-p}(\alpha)}\) |
To determine the F value, enter the equation FINV(\(\alpha,\text{num df, denom df})\) |
To determine the F- value, create F= finv(\(1 - \alpha, \text{p, n-p})\) |
Example 5-4 Section
This example uses the dataset that includes mineral content measurements at three different arm bone locations for n = 25 women. We’ll determine confidence intervals for the three different population means. Sample means and standard deviations for the three variables are:
Dataset: mineral.csv
| Variable | N | Mean | Std Dev |
|---|---|---|---|
| domradius | 25 | 0.84380 | 0.11402 |
| domhumerus | 25 | 1.79268 | 0.28347 |
| domulna | 25 | 0.70440 |
0.10756 |
Click to expand the solution using each method.
We’ll use a .95 confidence level for each interval. With n = 25, df = 24 and \(t _ { 24 } ( .025 ) = 2.064\). This can be found in Excel as =TINV(.05,24).
The confidence intervals have the form \(\bar{x}_j \pm 2.064\dfrac{s_j}{\sqrt{n}}\). Intervals are the following.
- For dominant radius:
\(0.84380 \pm 2.064 \dfrac{0.11402}{\sqrt{25}}\) which is 0.797 to 0.891
- For dominant humerus:
\(1.79268 \pm 2.064 \dfrac{0.28347}{\sqrt{25}}\) which is 1.676 to 1.910
- For dominant ulna:
\(0.70440 \pm 2.064\dfrac{0.10576}{\sqrt{25}}\) which is 0.660 to 0.749
We’ll use a .95 confidence family-wide level so the family error = .05. For each interval, the error rate = .05/3 = 0.016… The multiplier is \(t _ { 24 } ( .008333 ) = 2.574\) which can be found in Excel as =TINV(.05/3,24).
The confidence intervals have the form \(\bar{x}_j \pm 2.574\dfrac{s_j}{\sqrt{n}}\). Intervals are the following.
- For dominant radius:
\(0.84380 \pm 2.574 \dfrac{0.11402}{\sqrt{25}}\) which is 0.785 to 0.903
- For dominant humerus:
\(1.79268 \pm 2.574 \dfrac{0.28347}{\sqrt{25}}\) which is 1.647 to 1.939
- For dominant ulna:
\(0.70440 \pm 2.574 \dfrac{0.10576}{\sqrt{25}}\) which is 0.649 to 0.760
The necessary F value is \(\sqrt{\dfrac{3(25-1)}{25-3}3.049} = 3.159\). (See Example 3 above for details)
The confidence intervals have the form \(\bar{x}_j \pm 3.159 \dfrac{s_j}{\sqrt{n}}\). Intervals are the following.
- For dominant radius:
\(0.84380 \pm 3.159 \dfrac{0.11402}{\sqrt{25}}\) which is 0.772 to 0.916
- For dominant humerus:
\(1.79268 \pm 3.159 \dfrac{0.28347}{\sqrt{25}}\) which is 1.614to 1.972
- For dominant ulna:
\(0.70440 \pm 3.159 \dfrac{0.10576}{\sqrt{25}}\) which is 0.636 to 0.773
Steve Rathbun, formerly of Penn State, wrote the following SAS code (download below) to generate confidence intervals for population means using the three methods discussed here. The code reads a dataset, reshapes it to have a data line for each variable value, determines means and standard deviations, and then calculates and prints the three types of intervals. To use this code for different situations, you need only to change the third line where the value of p is set and the data step where the data set is read and reshaped.
- Dataset: mineral.csv
- Download the SAS program here: CI_pop_means.sas
The output for the program just given is below. It includes the sample mean and variance for each variable and the three confidence intervals. Limits for the one-at-a-time intervals are given as loone and upone. Limits for the Bonferroni method are given as lobon and upbon. Limits for the simultaneous confidence region method are given as losim and upsim.
| Obs | variable | _TYPE_ | _FREQ_ | n | xbar | s2 | t1 | tb | f | loone | upone | lobon | upbon | losim | upsim |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | domhumeru | 0 | 25 | 25 | 1.79268 | 0.080357 | 2.06390 | 2.57364 | 3.04912 | 1.67567 | 1.90969 | 1.64677 | 1.93859 | 1.61358 | 1.97178 |
| 2 | domradius | 0 | 25 | 25 | 0.84380 | 0.013002 | 2.06390 | 2.57364 | 3.04912 | 0.79673 | 0.89087 | 0.78511 | 0.90249 | 0.77176 | 0.91584 |
| 3 | dumulna | 0 | 25 | 25 | 0.70440 | 0.011568 | 2.06390 | 2.57364 | 3.04912 | 0.66000 | 0.74880 | 0.64904 | 0.75976 | 0.63645 | 0.77235 |
Walkthrough of the Three Methods using Minitab
To calculate 95% one-at-a-time confidence intervals:
- Open the ‘mineral’ data set in a new worksheet.
- Calc > Basic Statistics > 1-sample t
- Choose ‘One or more sample’ in the first window.
- Highlight and select ‘domradius’ and any other variables of interest to move them into the window on the right.
- Under ‘Options’, choose 95.0 for the confidence level and select Mean not equal to hypothesized mean.
- Select ‘OK’ twice. The one-at-a-time intervals are displayed in the results area.
To calculate simultaneous 95% confidence intervals via the Bonferroni method:
- Open the ‘mineral’ data set in a new worksheet.
- Calc > Basic Statistics > 1-sample t
- Choose ‘One or more sample’ in the first window.
- Highlight and select ‘domradius’, ‘domhumerus’, and ‘domulna’ to move them into the window on the right.
- Under ‘Options’, enter 0.9833, which corresponds to 1-0.05/3, the adjusted individual confidence level for simultaneous 95% confidence with the Bonferroni method.
- Select Mean not equal to hypothesized mean.
- Select ‘OK’ twice. The 95% Bonferroni intervals are displayed in the results area.
To calculate simultaneous 95% confidence intervals with the F-multipliers (based on the T-squared distribution):
- Open the ‘mineral’ data set in a new worksheet.
- Find the mean, standard deviation, and sample size needed for the calculations.
- Stat > Basic Statistics > Display Basic Statistics
- Highlight and select ‘domradius’, ‘domhumerus’, and ‘domulna’ to move them into the window on the right.
- Under ‘Statistics’, choose the mean, standard deviation, and ‘N nonmissing’.
- Select ‘OK’ twice. The statistics are displayed in the results area.
- Find the F-multiplier for simultaneous 95% confidence.
- Calc > Probability Distributions > Inverse Cumulative Distribution Function
- Choose ‘A single value’ and enter 0.95 in the ‘Value’ window.
- Select ‘F distribution’, and enter 3 (the number of variables) and 22 (sample size minus the number of variables) for the numerator and denominator degrees of freedom, respectively
- Select ‘Display a table of inverse cumulative probabilities’
- Select ‘OK’. The F-multiplier 3.049 is displayed in the results area.
- Create two new columns in the worksheet labeled ‘losim’ and ‘upsim’
- Calc > Calculator
- Highlight and select ‘losim’ to move it to the first window.
- In the Expression window, enter the formula 0.8438 - 0.1140/sqrt(25) * sqrt(3*24*3.049) for the lower confidence interval limit for domradius. Note that the values 0.8438, 0.1140, and 25 are the mean, standard deviation, and sample size obtained above.
- Select ‘OK’. The lower confidence limit is displayed in the worksheet under ‘losim’.
- Repeat sub-steps 3. and 4. above but use the formula 0.8438 + 0.1140/sqrt(25) * sqrt(3*24*3.049) for the upper limit, and choose ‘upsim’ for the result location.
- Repeat steps 4. and 5. above for domhumerus and domulna by substituting the corresponding means and standard deviations into the expressions.