The purpose of a one-sample t-test is to test a hypothesis involving the mean of a population against an Action Level. However, Barnard's sequential t-test deviates from standard t-test sample size equations. The investigator defines the number of samples per round (minimum of 10 samples in round one) and VSP iteratively uses the equations below to define when no further rounds of sampling are required.
According to Barnard's Sequential method, the number of samples is considered not adequate when:
Assume site is clean: \( \large log \frac{\beta}{1 - \alpha} \geq log \, L_n \geq log \frac{1 - \beta}{\alpha} \)
Assume site is dirty: \( \large log \frac{\alpha}{1 - \beta} \geq log \, L_n \geq log \frac{1 - \alpha}{\beta} \)
where:
\(log\) |
is the natural logarithm |
\(\alpha\) |
is the maximum Type I error rate |
\(\beta\) |
is the maximum Type II error rate |
\(L_n\) |
is the Likelihood ratio after \(n\) samples given by: when assume site is clean: \(L_n = \frac{pdft_{HA}}{pdft_{H0}} \) |
|
when assume site is dirty: \( L_n = \frac{pdft_{H0}}{pdft_{HA}} \) |
\( pdft_{H0} \) |
the Noncentral t probability density function with \(n - 1\) degrees of freedom and a non-centrality parameter of 0 |
\( pdft_{HA} \) |
is the Noncentral t probability density function with \( n - 1\) degrees of freedom and a non-centrality parameter of \(\delta\) |
\(\delta\) |
is the non-centrality parameter given by \( \frac{\Delta}{Err_{std}} \) |
\( \Delta\) |
is the width of the gray region given by \(HA-H0\) (note: when the site is assumed dirty, \(\Delta\) is negative) |
\(Err_{std}\) |
is the standard error given by \( \sqrt{ \frac{ \sum_{i=1}^n (x_i - u )^2}{n(n-1)}} \) (standard deviation of the means) |
\(u\) |
is the mean of the sample values given by \( \frac{ \displaystyle\sum_{i=1}^n X_i}{n} \) |
\(x_i\) |
is the \(i^{th}\) sample value |
\(n\) |
is the number of sample values |
Special note: when calculating the mean and standard error for \( n \) < 10, VSP uses all of the first 10 values (if available).
The assumptions associated with the sequential sampling formulas are:
The data are sampled sequentially from a normal distribution with an unknown mean \(\mu\) ,
The variance estimate, \(s^2\), is reasonable and representative of the population being sampled,
The population values are not spatially or temporally correlated, and
The sampling locations will be selected probabilistically.
The first three assumptions will be assessed in a post data collection analysis. The last assumption is valid because the sample locations were selected using a random process.
Other sampling designs (like the one sample t-test) calculate the number of samples needed to meet the sampling objectives, place that number of sampling locations on the map and then leave it up to the user to follow up appropriately. However, the sequential designs are by their nature iterative, requiring the user to take a few samples and enter the results into the program before determining whether further sampling is necessary to meet the sampling objectives.
Barnard's sequential design in VSP works as follows (this assumes you are using a map with at least one selected sample area):
1. After opening the design dialog, input the test parameters like alpha, beta, action level, etc. Next, input the Number of Samples Per Round. This parameter indicates how many samples you want to take each time you mobilize out into the field. Each time you press the Apply button, VSP places a pattern of sampling locations on the map. (VSP requires a minimum of 10 samples before making a decision. If you specify fewer than 10 samples per round, VSP will place 10 sampling locations in the initial round.)
2. After collecting the samples or otherwise determining the values at the specified location, enter the values in the Data Analysis / Data Entry page. Upon returning to the Mean vs. Fixed Threshold page, VSP will determine whether more samples are needed to accept or reject the null hypothesis.
3. If it is determined that more samples are needed before making a decision, VSP will tell you take \(N\) more samples, where \(N\) is the number of samples per round that you entered. Each time you press the Apply button, VSP will place a pattern of sampling locations on the map (avoiding existing sampling locations).
Steps 2 and 3 are repeated until there is enough data to either accept or reject the null hypothesis.
It is useful to watch the Barnard's Sequential graph during the process to see how close VSP is to making a decision. The graph can be viewed on the Data Analysis / Plots page or on the Graph View. There are 2 versions of the graph you may use for Barnard's Sequential t-Test: plot of the data mean or the plot of the test statistic (which is the log likelihood ratio). You can change the graph view by choosing Options / Graphs / Barnard's Log Likelihood Ratio from the menu or by clicking on the Y-Axis label of the graph.
Version 2.0 Visual Sample Plan (VSP): Models and Code Verification. PNNL-13991, Pacific Northwest National Laboratory, Richland, Washington. Gilbert, RO, JE Wilson, RF O'Brien, DK Carlson, DJ Bates, BA Pulsipher, CA McKinstry.
G. A. BARNARD: The Frequency Justification of Certain Sequential Tests. Biometrika 1952 39: 144-150.
Width of Gray Area (Delta) / LBGR / UBGR (when null hypothesis="site is unacceptable")
Width of Gray Area (Delta) / LBGR / UBGR (when null hypothesis="site is acceptable")
Type II Error Rate (Beta) (when null hypothesis="site is unacceptable")
Type II Error Rate (Beta) (when null hypothesis="site is acceptable")
Data Analysis page