EJC T TEST

1 EJC H2 Biology T2W4 Biostatistic – t test Student’s t test 1. Introduction The main reason for carrying out observations and experiments in Biology is to test an idea or hypothesis; Biology, as a science, is an on-going subject, answers to one question, usually leading to further questions. Such work requires data sets. What we want to know is whether the data supports the initial hypothesis or not; does the data en able us to accept our hypothesis as a valid explanation or do we have to modify the hypothesis, or even reject it? You have already been introduced to one statistic al test that is used to analyse data sets – the ch i- squared test or Χ2 test. We will look at another test – the t test, in this lecture set. 2. Learning Outcomes (a) Able to apply t-test in Biological situations (b) Able to calculate standard deviations (c) Able to test for significant differences between means of two small unpaired samples Use the knowledge gained in this section in new situations or to solve related problems 3. References Gavin, J.W., Skills in Advanced Biology – Dealing with Data I Contents 1. Introduction ............................................................................................................... 1 2. Learning Outcomes.................................................................................................... 1 3. References ................................................................................................................. 1 4. Tests of Significance .................................................................................................. 2 5. Null Hypothesis, H0 .................................................................................................... 2 6. Χ2 test vs t test…………………………..………………………………………………………………………….2 7. Normal Distribution Curve……………………………………………………………………………..………3 Standard Deviation………………………………………………………………………………………….4 8. History of Student's t test………………………………………………………………………………………4 9. Calculating the t value..….……………………………………………………………………………………..6 Barnard, C., Gilbert, F. & McGregpr, P. - Asking Questions in Biology, 2nd Edition. T Test as a Parametric statistic - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4667138/ Bread and Butter of Statistical Analysis: T Test - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4744321/ Statistics: A Brief Overview - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3096219/

2 EJC H2 Biology T2W4 Biostatistic – t test 4. Tests of Significance Biological data is variable. It consists of small samples taken from large populations. Even though these samples are randomly chosen it is very unlikely that the data will conform exactly to the hypothesis. To overcome this problem, tests of significance are used to determine whether the data conform to the hypothesis sufficiently closely for a decision to be made, not only about the samples but also about the general populations from which they were drawn. Often the hypothesis requires a comparison between two or more samples. The degree of difference between samples may be very small, very large, or anything in between. The real purpose of the exercise is to find out, from the difference between the samples, whether they were drawn from the same (statistical) population or from different (statistical) populations. We cannot be certain about any statistical conclusion – a statistical test does not prove anything. It does, however, give us some degree of confidence in our conclusions. 5. Null Hypothesis, H0 This is a ‘negative’ hypothesis: for example, if we were comparing two samples, the null hypothesis would be stated in the form “there is NO significant difference between the two samples”. It assumes that any differences that do occur are due to chance. The significance test then confirms whether to not reject or to reject, the null hypothesis. If the null hypothe sis is not rejected – then there is NO significant difference between the two samples. If the null hypothesis is rejected – then there is a significant difference between the two samples. 6. Χ2 test vs t test A Χ2 test can be used if data are in the form of counts i.e. two groups have been identified and observations classified in terms of the number belonging to each. Χ2 can only be used on raw counts; it cannot be used on measurements (e.g. length, time, weight, volume etc) or proportions, percentages or any derived values. The test works by comparing observed counts with those expected by chance or a prior basis. Experi ments in Mendelian genetics would be such an example. Sizes of samples in a Χ2 test are generally are over 30. The t test can be performed on raw counts like Χ 2 test but it deals with each contributing data value in the two groups separately instead of as a single total. It can deal with data other than counts e.g. body size, time spent performing a certain behaviour, percentage of patients responding to a drug etc. Sample sizes can be around 30 (or sometimes less). Notes to self

3 EJC H2 Biology T2W4 Biostatistic – t test 7. Normal Distribution Curve Many if not most characteristics in Biology (e.g. heights of people, weight of rats, number of flowers in plants, number of peas in a pod, body temperature of mice, breathing rate of locusts etc) fall into a normal distribution ( refer to Continuous Variation in Genetics). When drawn as a graph, the normal distribution assumes a characteristic ‘bell’ shape. A normal distribution is symmetrical about the mean value with 50% of the values less than the mean and 50% of the values more than the mean. The mean = median = mode. Standard Deviation The standard deviation (SD) is a measure of the spread of data values from the mean (assuming the data follow normal distribution) . It is a measure of the confidence you have that any particular data value will fall within a particular range (the mean + SD and the mean – SD). In a normal distribution, 68% of the values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean.

4 EJC H2 Biology T2W4 Biostatistic – t test Not in syllabus (but you will see this in data presented in many scientific papers) The standard error of the mean (SE) measures the spread of multiple sample means around the true population mean. The assumption is that when we take a very large number of random samples from a population, the means from each sample will form a normal distribution. The distribution of sample means will have a mean which is close to actual population mean. Normally we can only take data from a few samples and not a large number of samples, so the SE is an expression of the confidence we have that our sample means falls within a particular range (mean ± SE) of the true population mean. Since sample means can be conside red to be normally distributed, we can cite the standard error with sample mean. Calculating the Standard Deviation (SD) Suppose we have a set of data where we measured the body lengths of ten grasshoppers caught in a geographical area and obtained following results: Length, x (cm) 6.3 7.1 6.2 6.5 7.0 6.7 6.5 7.0 6.8 7.1 Mean x̅ = 6.7 n (sample size) = 10 Standard deviation has the formula: s = √∑(𝑥−𝑥̅)2 𝑛−1 Note: s may be denoted as  in calculators Let’s work out the SD from the data above: Length, x (cm) x2 (x - x̅ ) (x - x̅ )2 6.3 39.69 - 0.4 0.16 7.1 50.41 0.4 0.16 6.2 38.44 - 0.5 0.25 6.5 42.25 - 0.2 0.04 7.0 49.00 0.3 0.09 6.7 44.89 0.0 0.00 6.5 42.25 - 0.2 0.04 7.0 49.00 0.3 0.09 6.8 46.24 0.1 0.01 7.1 50.41 0.4 0.16 Mean, x̅ = 6.72 x2 = 452.58 (x - x̅ )2 = 1 x = 67.2 (x)2/n = 451.58 n (sample si