Hypothesis Testing - the scientists' moral imperative

1. Hypothesis Testing - the scientists' moral imperative • To tell whether our data supports or rejects our ideas, we use statistical hypothesis testing. • The problem is that we often get data that seem to support our ideas. The literature is full of papers that accept a pet idea uncritically. Statistical testing keeps scientists honest. • If you read a paper that suggests some alternative hypothesis should be accepted, but there is no statistical test, don't believe it. Immanuel Kant

2. If a topic is part of science, ideas have consequences that can be checked. We can count and measure aspects of nature to check our ideas. If our measurements are contrary to the predictions of our model, our hypothesis is FALSE, and we reject it "It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is-- if it disagrees with experiment it is wrong. That is all there is to it." -Richard Feynman (The Character of Physical Law) http://video.google.com/videoplay?docid=-5157969812375041230&hl=en “If you haven't measured it you don't know what you are talking about.” -William Thompson, Lord Kelvin

3. The formula for finding the mean is: Mean = ∑x n The Mean • The mean is one of the commonly used statistics in science. It is often the "Expected Value" i.e. the value we expect to get. • The mean is found by totalling the values for all observations (∑x) and dividing by the total number of observations (n).

4. Standard Deviation • Measures of the spread of data are the standard deviation, s, andvariance s2 Sample variance • For s.d. calculate the difference of each observation and the mean, square it, add all these up, divide by the sample size, and take the square root. • Refers to actual population

5. Sample Standard Deviation • "It is rarely possible to obtain observations from every item … in a population" Fowler et al. 1998 p36 • The estimate of s is s • Where N-1 is called the degrees of freedom, and is one less than the number of observations

6. Knowing the Distribution • In coin toss experiments, we know a formula for calculating the probability of any number of k heads in n trials, the Binomial Distribution. • Fortunately, we don’t have to know the distribution for every situation in nature. • We are saved by the Central Limit Theorem

7. Central Limit Theorem • “… the means of a large number of samples drawn randomly from the same population are normally distributed ….".Fowler et al. 1998 p 91 • So it “pays us” to use means, not raw data Means distributed like this, formula known Sample this, formula unknown

8. Normal Distribution • So if we make many observations and use averages as our data, we can draw valid conclusions because we know their distribution • Many test statistics are available for this. • In a few moments we will learn Chi-Square (X2)

9. Expected Value • The expected value is the average result we expect. • It is the product of the probability times the number of observations E = p x n • A really useful case is the where the probability of all cases is equal. • For example, in fair coin tosses, the probability of Heads =1/2. If we flip a coin 24 times, we EXPECT ½ x 24 = 12 Heads • Equal probability cases are usually the basis of the "Null Hypothesis"

10. Hypotheses • A hypothesis is a statement of the researcher’s idea or guess. • To test a hypothesis the first thing we do is write down a statement – called the null hypothesis. • The null hypothesis is often the opposite of the researcher’s guess.

11. For Example • Some null hypotheses may be: • “there is no difference in lava viscosity between Hawaiian and Cascades volcanoes”. • “there is no relation between a volcanic islands’ height and its age.” • “there is no connection between the time since subaerial exposure of sediment and the height of the hills it forms”

12. The Hypotheses • Null Hypothesis H0: ‘There is no differencebetween the average number of streams per square kilometer and the bedrock type.' • Alternative Hypothesis HA or H1: ‘There is a differencebetween the average number of streams per square kilometer and the bedrock type.'

13. Significance (1) • Before carrying out any test we have to decide on a significance level which lets us determine at what point to reject the null hypothesis and accept the alternative hypothesis.

14. Significance (2) • Significance is based on the probability of a particular result. • Statisticians have calculated the probability of all possible ‘chance’ events occurring.

15. Significance (3) • Many trials, and the use of a higher significance level (P=.01 not P=.05) , make this less likely • If the probability of a particular result is less than 1 in 20 (P=0.05), we say the result is significant, ie: the result is not just a chance event. • If the probability of a particular result is less than 1 in 100 (P=0.01), we say the result is highlysignificant; again, the result is not just a chance event.

16. Avoiding Decision Errors • We always run the risk that we will observe a rare event, and we will draw the wrong conclusion. • Usually we want to avoid a Type I error, where we reject H0 even though it is true. • Many trials, and the use of a higher significance level (P=.01 not P=.05) , make this less likely • In a Type II error, we accept the null hypothesis even though it is false. • We will see an example of a type II error later.

17. Test Statistics • We often want to see if two things are different from each other. • In science we calculate what we would expect if there was no difference between them (the usual null hypothesis). • We can then compare this to what we actually observe.

18. Test Statistics • To check the null hypothesis we calculate a figure known as a test statistic, which is based on data from our samples. • Different types of problems require different test statistics. Values for comparison to our data have all been put into statistical tables. • All we need to do is to calculate our value and compare it with the value in the table to get our answer.

19. Using a Test Statistic If the test statistic shows you “observed an unlikely result”, you reject the null hypothesis and accept the alternative hypothesis

20. The significance levels available on a x2 table are usually 0.05, 0.01, and .001, which means there is, respectively, only a 1 in 20 (0.05), a 1 in 100 (0.01), or a 1 in 1000 (0.001), probability of the event occurring by chance if that x2 is obtained. The values in the tables are called critical values. x2 Significance Tables

21. Chi-Square (X2)Critical Values In Use • For Chi-Square (X2) : If the value of the test statistic you have calculated is greater than the value in the table (the critical value) you decided to use, you can reject the null hypothesis and accept the alternative hypothesis. H0: There is no relation between the number of peaks along a ridge and the time since exposure

22. Chi-Square (X2) Critical Values Significance table of X2 values. This is the critical value table we will use in the examples below.

23. Important It is prudent to use means, not raw data, to insure a normal distribution • The chi square test can only be used on observations that have the following characteristics: Objects being counted are independent** The frequency data must have a precise numerical value and must be organised into categories or groups. The data must be in the form of frequencies The expected frequency in any one cell of the table must be greater than 5. * The total number of observations must be greater than 20. * See the exception next slide **There are statistics designed to test this assumption

24. An Exception The expected frequency in any one cell of the table must be greater than 5. • "The discrepancy is not large, however, when X2is computed from contingency tables with a fairly large number of cells (more than 4, at a minimum) and only a few theoretical frequencies are less than 5." • Source: Spence, J. et al.(1968) Elementary Statistics

25. Other Statistics • If any of the assumptions for X 2 are false, we cannot use X 2 • However, there are test statistics for most situations, and they are all similar in their use. • Once you know X 2 you can look up the correct statistic and apply it

26. The x2formula Smeans take the sum

27. Worked Example 1: • Step 1. Write down the NULL HYPOTHESIS (H0) and ALTERNATIVE HYPOTHESES (Ha) and set the LEVEL OF SIGNIFICANCE. • H0'A basaltic sand pile will not spread further than a quartz sand pile in the same time' • Ha' A basaltic sand pile will spread further than a quartz sand pile in the same time ' • We will set the level of significance at 0.05.

28. Step 2: Construct a table with the information you have observed. Use averages as data Method: 220 Hawaiian sand and 250 New Jersey sand piles of 50 cc each are left out in the weather for 1 week After 1 week, the distance of the furthest grain beyond the initial perimeter is measured. Every 5 piles are averaged. Note that although there are 3 cells in the table that are not greater than 5, these are observed frequencies. It is only the expected frequencies that have to be greater than 5.

29. Expected frequency = row total x column total Grand total Work out the expected frequency. Eg: expected frequency for oaks in PL1 = (50 x 13) / 92 = 7.07 You do the rest

30. The Expected Frequencies

31. (O – E)2 E For each of the cells calculate: Eg: Basaltic in 1-5 mm is (9 – 7.07)2 / 7.07 = 0.53 You do the rest

32. (O – E)2 E These are the x2 = (O – E) 2 E But S So: Add up all of the above numbers to obtain the value for chi square: x2 = 15.14.

33. Now: • Look up the X2 value on the table in the slide above. This will tell you whether to accept the null hypothesis or reject it. The number of degrees of freedom to use is: the number of rows in the table minus 1, multiplied by the number of columns minus 1. This is (2-1) x (5-1) = 1 x 4 = 4 degrees of freedom. We find that our answer of 15.14 is greaterthan the critical value of 9.49 (for 4 degrees of freedom and a significance level of 0.05) and so we reject the null hypothesis.

34. We conclude: ‘The distribution of grains spreading from sand piles made of basaltic minerals versus quartz is significantly different.’ Now you have to look for physical factors to explain your findings If you ask me, you should check the density. Basaltic Hawaiian basaltic sands are mostly pyroxenes The density of pyroxene is 3.24 g/cm3. The density of quartz is 2.536 The pyroxene has greater potential energy when hilled to the same height as quartz sand