HYPOTHESIS TESTING

1. HYPOTHESIS TESTING

2. Introduction In making inference from data analysed, there is the need to subject the results to some rigour. Drawing meanings from data in this way is called Inferential Statistics Hypothesis testing is one of the important tools in inferential statistics.

3. Introduction The issues to be considered involves: identifying the null and alternative hypotheses deciding on an appropriate significance level issues to do with Type I and Type II errors, one or two-tailed tests, power. existence of multiple tests of significance.

4. Introduction A hypothesis is a conjecture In the sciences, the hypothesis is the beginning of a theory The hypothesis is formulated , tested and if acceptable the, becomes a theory.

5. Types of Hypothesis We need to distinguish between a research hypothesis and a statistical hypothesis Research hypothesis does not require statistical tests to validate Statistical hypothesis requires statistical tests to validate

6. Statistical Hypothesis There are generally two forms of a statistical hypothesis: null (typically represented as H0 [pronounced "H naught�]); alternative (typically symbolised as H1or Ha- the one we are really interested in showing support for).

7. Example of Statistical Hypotheses

8. hypotheses are usually formally stated in terms of the population parameters about which the inference is to be made. Because our interest is in making an inference from sample information to population parameter(s),

9. We use two forms of hypotheses to set the stage for a logical decision. If we amass enough evidence to reject one hypothesis, the only other state of affairs which can exist is covered by the remaining hypothesis. Thus, the two hypotheses (null and alternative) are set up to be mutually exclusive and exhaustive of the possibilities.

10. Direction of Hypotheses Directional (one tail) Non directional (two tail test) Examples:

11. Direction of Hypotheses Directional Also termed �one tail test� Use < or > where the direction of deviation from the null value is clearly specified; a specific predicted outcome is stated. one-tailed tests, should only be used in the light of strong previous research, theoretical, or logical considerations. Hence a "claim", "belief", or "hypothesis", or "it was predicted that" is not sufficient to justify the use of a one-tailed test.

12. Direction of Hypotheses- One tail A directional hypothesis only considers one tail (the other tail is ignored as irrelevant to H1), thus all of can be placed in that one tail.

13. Direction of Hypotheses- Non-Directional Non Directional Also termed �two tailed test� Use the direction of deviation of the alternative case is not specified A two-tailed test requires us to consider both sides of the normal distribution, so we split and place half in each tail.

14. Level of Significance Also known as alpha ( ) level specifies the probability level for the evidence to be an unreasonable estimate Unreasonable means that the estimate should not have taken its particular value unless some non-chance factor(s) had operated to alter the nature of the sample such that it was no longer representative of the population of interest. The researcher has complete control over the value of this significance level.

15. The choice is for levels up to 10% in social sciences The lower the level of significance the stronger the effect or phenomenon being studied. The -level should be considered in light of the research context and in light of your own personal convictions about how strong you want the evidence to be, before you will conclude that a particular estimate is reasonable or unreasonable.

16. In some exploratory contexts (perhaps, in an area where little previous research has been done), you might be willing to be more liberal with a decision criterion and relax the level of significance to 0.10 or even 0.20. Thus, less extreme values of a statistic would be required for in order to conclude that non-chance factors had operated to alter the nature of the sample.

17. On the other hand, there are research contexts in which one would want to be more conservative and more certain that an unreasonable estimate had been found. In these cases, the significance level might be lowered to 0.001 (0.1%) where more extreme values of a statistic would be required before non-chance factors were suspected

18. SPSS output usually under heads it as "Sig." or "Two-tailed Sig.", or "Prob." for the probability (or "p-value") of your results being a real difference or a real relationship. This can then be the probability you quote as being the "level of significance" associated with your results. Note that it is normally required that this p-value to be less than or equal to 0.05 to make the claim of �significant�.

19. It is then up to your discussion to explain/justify/interpret this level of significance to your reader As a general guide, treat the p-value as a measure of the confidence or faith in the results being real (and not being due to chance fluctuations in sampling).

20. The -level is the probability or �p-value� you are willing to accept as significant. Ideally, this -level. The -level can also be interpreted as the chance of making a Type I error.

21. Type I Error and Type II Error When alpha is set at a specified level (say, 0.05) it indicates automatically specify how much confidence (0.95) is placed in the decision to "fail to reject Ho if it really is the true state of affairs.

22. Type I Error and Type II Error Consider doing the same experiment exactly 100 times, each time using a different random sample If alpha is set to 0.05 (and consequently 1 - = 0.95), then in the 100 experiments, it should be expected to make an incorrect decision in 0.05 x 100 or 5 of these experiments (= 5% chance of error), and a correct one 95% of the time if Ho is really true.

23. Type I Error and Type II Error

24. Type I Error and Type II Error Bear in mind that you are merely making a decision concerning what you believe about the truth or falsity of the hypothesis; you are not really ascertaining whether the hypothesis is true or false. In other words, if you decide to reject H0, that means "I have decided to believe that H0 is false"; it does not necessarily mean that H0 is actually false.

25. Type I Error and Type II Error Similarly, if the decision is to accept (or, more precisely, not to reject) H0, that means "I have decided to believe that H0 is true"; it does not necessarily mean that H0 is actually true.

26. Type I Error and Type II Error Thus, states what chance of making an error (by falsely concluding that the null hypothesis should be rejected) we, as researchers, are willing to tolerate in the particular research context.

27. Type I Error and Type II Error When we consider the case where Ho is not the true state of affairs in the population (i.e., Ho is false), we move into an area of statistics concerned with the power of a statistical test. If Ho is false, we want to have a reasonable chance of detecting it using our sample information.

28. Type I Error and Type II Error Of course, there is always the chance that we would fail to detect a false Ho, which yields the Type II or error However, error is generally considered less severe or costly than an error We must be aware of the power of a statistical test �the test's ability to detect a false null hypothesis� because we want to reject Ho if it really should be rejected in favour of H1

29. Type I Error and Type II Error Hence we focus on 1 - alpha which is the probability of correctly rejecting Ho.

30. Type I Error and Type II Error It can be helpful to think of the Type I error as "rejecting the null hypothesis when it is true" and the Type II error as "failing to recognize that the null hypothesis is false when it is false� Another way of putting it is that the Type I error amounts to "disbelieving the truth"; the Type II error, to "believing an untruth�

31. Power of Test Let�s depict a hypothesis testing situation graphically using two normal distributions one to represent the sampling distribution for the null hypothesis the other to represent the sampling distribution for the alternative hypothesis the sampling distribution of the mean from any population looks more and more normal as sample size, N, is increased � central limit theorem; that is why the normal distribution is used.

32. Power of Test Thus, if a reasonable sample size, is assumed it is justified to use normal distributions to represent the situation. In this illustration we are using sampling distributions (in this case for the sample mean) as standards against which to compare our computed sample statistic.

33. In general terms, suppose we evaluate the specific hypothesis that � has one value (� Ho) or another value (� H1) using a sample of size N: Ho: � = �Ho versus H1: � = �H1;

34. Power of Test

35. Power of Test Four factors interact when we consider setting significance levels and power: 1. Power: 1- (probability of correctly concluding "Reject Ho") 2. Significance level: (probability of falsely concluding "Reject Ho") 3. Sample size: N 4. Effect size: e (the separation between the null hypothesis value and a particular value specified for the alternative hypothesis).

36. One way to increase power is to relax the significance level ( ) [if e and N remain constant]

37. Another way to increase power is to increase the sample size

38. Yet another way to increase power is to look only for a larger effect size [if and N remain constant].

39. Finally, power can be increased if a directional hypothesis can be stated (based on previous research findings or deductions from theory).