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Exercise 1

This exercise is based on research carried out by an undergraduate. The topics covered are those in 4.1 and appendix b.

Example W4.1: Microbial counts of prepared sandwiches

An undergraduate investigated the response of pre-prepared sandwiches to storage temperature. Three types of pre-prepared sandwiches with the same sell by date were purchased, 21 of each type, and placed immediately into a cool bag for transport to the laboratories. On arrival all sandwiches were placed at room temperature. A sample was taken from each type of sandwich at the following time points: 0 mins, 15 mins, 30 mins and 60 mins, 75 mins, 90 mins and 120 mins. At each time point a fresh sandwich of each variety was opened and sampled. There were three replicates for each type of sandwich for each time point. 10 grams of sandwich were taken aseptically at the given time point and placed in a stomacher where it was mixed with a diluting solution. After 90 seconds, 1cm3 of the suspension was serially diluted and 1cm2 of the dilution used to inoculate an agar plate. The plates were sealed and incubated for 24 hours at 37°c after which time the number of colonies on each plate were recorded.

1
Q W1.1

Which statistical test(s) should be suitable for analysing these data?

[If you would like to save a record of your answer, please type it into this Word document instead of the text box below]

A two-way non-parametric ANOVA.

To come to this conclusion you should follow the steps outlined in appendix b.

B.1 What type of investigation am I designing?

This is an experiment, you are starting out with a question (hypothesis) (go to B.2).

B.2 Which type of hypotheses am I testing?

There are three types of hypotheses which you need to choose between. If you are not sure which type of hypotheses you will be testing read the information in B2.1 - B.2.3 before deciding. For more information about hypotheses and hypothesis testing read Chapter 4.

In our example the student does not have an expectation. Since he was comparing two treatment variables (type of sandwich and storage time) he was not sure if he wanted to test for an association or examine a difference. Having looked at the information about the statistical tests for testing these hypotheses he decided to test for differences (Hypotheses type 3).

B.2.3 Do samples come from the same or different populations?

These are the type of hypotheses that are most frequently investigated by undergraduates. There are many tests that will test this type of hypotheses. These tests fall into parametric tests (Chapter 7) to be used when you have Normally distributed data and non-parametric tests (Chapter 8) when your data are not Normally distributed. To tell if your data are Normally distributed refer to BOX 3.2.

In this example the observations recorded will be 'numbers of bacteria'. This is an ordinal, non-parametric measure. Therefore we should consider non-parametric tests.

B.2.3.2 Non-parametric tests

The choice of these tests depends on how many treatment variables you are planning to examine, how many categories in each variable, and how many replicates in each category.

In our example there are two variables (storage time and type of sandwich). The first has 7 categories/samples and the second has 3 categories/samples. There are three replicates for each category. From the table it appears that a non-parametric ANOVA is probably the correct test to use for these data.

Experimental design

Test

You have one treatment variable. You are going to compare two samples. The data is unmatched. You have 20 observations or less in each sample.

Mann Whitney U test (8.1.)

You have one treatment variable. You are going to compare two samples. The data is unmatched. The data is measured on a continuous scale and you have more than 30 observations in each sample.

z test for unmatched data (7.1.)

You have one treatment variable. You are going to compare two samples. The data is unmatched. You have more than 20 observations in each sample.

Sokal & Rohlf, 1981.

You have one treatment variable. You are going to compare two samples. The data is matched. You have less than 30 pairs of observations.

Wilcoxon's rank paired test (8.2.)

You have one treatment variable. You are going to compare two samples. The data is matched. You have more than 30 pairs of observations.

z test for matched data (Chapter 7 (7.2).

You have one treatment variable. You are going to compare two or more samples. You wish to test general and specific hypotheses.

One-way ANOVA (Kruskal Wallis test)( 8.3. and 8.4)

You have more than one treatment variable. You are going to compare two or more samples. You wish to test general and specific hypotheses. You will be using a calculator.

Two-way non parametric ANOVA (8.5. and 8.6)

You have more than one treatment variable. You are going to compare two or more samples. You wish to test general hypotheses. You want to use a computer.

Scheirer-Ray-Hare test (8.7).



The criteria for using a non-parametric ANOVA (8.5.1) are that you:

  1. Wish to test for differences in population medians.
  2. Have two treatment variables each with at least two categories.
  3. Have an orthogonal design
  4. Have non-parametric data that can be ranked.
  5. Can test both general and specific predictions if there are equal numbers of observations in each sample. (If there are not equal numbers of observations in each sample only specific predictions can be tested).

All these criteria are met by the current design. Therefore, this appears to be a suitable statistical test to use to analyse the data.

Check your answer

2
Q W1.2

Write hypotheses for these statistical tests of the data.

[If you would like to save a record of your answer, please type it into this Word document instead of the text box below]

With the current design it will be possible to test both general and specific hypotheses. For the two-way non-parametric ANOVA you may test three pairs of general hypotheses. The specific hypotheses will be determined by the data (8.6).

The general hypotheses will be:

H0: There is no difference between the median number of bacterial colonies from three types of pre-prepared sandwiches.

H1: There is a difference between the median number of bacterial colonies from three types of pre-prepared sandwiches.

H0: There is no difference in the median number of bacterial colonies from samples taken after different times of storage (mins) at room temperature.

H1: There is a difference in the median number of bacterial colonies from samples taken after different times of storage (mins) at room temperature.

H0: There is no interaction between the type of pre-prepared sandwich and time in storage (mins) in the median number of bacterial colonies cultured.

H1: There is an interaction between the type of pre-prepared sandwich and time in storage (mins) in the median number of bacterial colonies cultured.

Check your answer