Sample size for hypothesis testing for the ratio of two standard deviations

The BIS.Net Team BIS.Net Team

The theory behind hypothesis testing for the ratio of two standard deviations is based on the theory for hypothesis testing of two variances where the variance is equal to the standard deviation ^2.

Just as an analyst can be interested in the difference between two averages, an analyst can also be interested in the difference between two standard deviations.

For example, how well a viscous glucose solution is mixed in the confectionery industry can be measured with the standard deviation of glucose content from multiple samples taken from the same vat. By comparing the standard deviation of two mixing processes after mixing is deemed to be completed the analyst can determine if one mixing process is better than another.

Similarly, blood glucose management can be evaluated by comparing blood glucose standard deviations before management with after blood glucose management.

The ratio of two standard deviations Sd1/Sd2 (F [Sd]) is a convenient way of comparing the two standard deviations. If there is no difference, then the ratio should be equal to 1+/-delta depending on sampling error.

To determine the sample size the analyst will need to specify a reference value for the null hypothesis, which can be considered the status quo. For this type of test the null hypothesis F (Sd) ratio is usually set to 1 meaning no difference in the two standard deviations. The amount of concern, i.e. the minimum change from the reference value that is important to detect must also be specified.

This is explained through an example. For example, consider the above process. If the standard deviation of the current mixing process is 2, and the Quality Manager will consider a new mixing process only if the standard deviation of the new process is equal to 1 (i.e. the standard deviation has reduced by a half of the original value), then the Null Hypothesis ratio is set to 1 and the amount of concern is also set to 1 (by coincidence). Why is now explained.

The key to understanding is knowing that this test is about ratios. The null hypothesis in this instance is that there is no change. If there is no change then the standard deviation of the alternative mixing processes is also equal to 2 and hence the null hypothesis ratio is equal to 2/2=1 as set above. The alternative hypothesis is equal to the standard deviation of the current process divided by the standard deviation of the new process i.e. 2/1=2. The amount of concern is therefore 2-1=1 as set above.

Additionally, the analyst must specify two types of risk. The first is the risk or probability of concluding there is a change in F (Sd) ratio at least equal to the specified amount of concern when there was none. This is called the alpha risk. The second is the risk or probability of concluding there has been no change in the population F (Sd) ratio when in fact there was a change. This is called the beta risk. Both risks are due to ‘the way the numbers fall’ when sampling. By chance sampling may have selected samples with measurements larger or smaller than expected. Finally, the type of change must be specified, which can be > or < or <>.

To determine the required sample size is however not possible through numerical methods.

Hypothesis testing on the ratio of two standard deviations

The calculated sample size must be such that the specified value of alpha and beta (such as shown above for a > than test) is obtained. Unfortunately for the F (Sd) ratio, where the probability distribution depends on the F (Sd) ratio itself, there is no closed form expression that can be used to directly or numerically calculate the sample size such that both risks are kept.

Instead a machine learning algorithm is used which learns to discard incorrect solution paths, ultimately finding the required sample size.

There are some limitations however. These are:

  • The underlying distributions must be normally distributed.
  • The actual risks only approximate the specified risks

There are no other known alternatives that our researchers are aware of. The machine powered algorithm provides the best alternative to estimating a sample size. The alternative is to take a stab.

Once the sample size is determined it is possible to calculate critical values for the F (Sd) ratio. An amount beyond the critical value implies that there is reasonable evidence that there has been a change. For a > than hypothesis an upper critical value is calculated. For a < than hypothesis a lower critical value is calculated. For a <> equal hypothesis both a lower and upper critical value is calculated.

A preferred and recommended option to using critical values is to perform a hypothesis test after sampling with the calculated sample size and make decisions based on the p value.

The BIS.Net Sample Size app also displays an OC and Power curve such as shown in the image below.

Hypothesis testing on the ratio of two standard deviations

The image applies to the above mixing process example with a specified alpha risk of .05 and beta risk of .1 (power of .9) and a computed sample size of 73.

The analyst can use this practical tool to answer what if questions, such as what will the power be if the alternative F (Sd) ratio is equal to more or less than the specified value (based on the amount of concern)

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