HYPOTHESIS TESTING Asst Prof Dr. Ahmed Sameer Alnuaimi

Learning objectives Define research hypothesis. Define statistical hypotheses. State the purpose of hypothesis testing. Value the difference between parametric and non-parametric statistical hypotheses. Define “Test Statistics”. Learn the formula for calculating “Test Statistic” in case of single proportion, difference between 2 proportions, single mean (under the condition of known and unknown population variance), difference between 2 means (under the condition of known and unknown population variance, with assumption of equal and unequal population variances)

Learning objectives Define type-I and type-II errors in hypothesis testing. Define P value. Define rejection region and acceptance region. Master the 8 steps of hypothesis testing procedure. Master the interpretation of hypothesis testing procedure.

Research Hypothesis It is the assumption that motivate the research. It is usually the result of long observation by the researcher. It is stated in a direct type of language. For example: “Is there a difference in school achievement between males and females”. This type of hypothesis lead directly to the second type of hypothesis, called Statistical Hypothesis.

Hypothesis Testing Purpose The purpose of hypothesis testing is to help the researcher or administrator in reaching a decision concerning a population by examining a sample from that population. Definition of Statistical Hypothesis It is a statement about one or more population. Usually concerned with the parameter of the population about which the statement is made (parametric hypothesis is used with tests based on normally distributed data). An exception is noticed with non-parametric (distribution free) tests (like Chi-square test and Mann-Whitney test), in which no parameters are mentioned in the hypothesis statement

Statistical Hypothesis It is stated in a way that can be evaluated by appropriate statistical techniques. It is composed of two types: Null hypothesis( Ho): It is the particular hypothesis under test. It is the hypothesis of “no difference or no association”. Alternative hypothesis (HA): which disagrees with the null hypothesis (and usually agrees with the research hypothesis).

Test Statistic It is a mathematical expression, which provides a basis for testing a statistical hypothesis . The result of this test will determine whether we will accept the Null hypothesis and as a results reject the HA (i.e. the researcher will be disappointed, because his theory does not hold and can not be generalized from the sample to the population). Or we reject the null hypothesis and so the HA will be accepted (i.e. the researcher will happily generalize the findings of his sample to the reference population).

Errors There are two possible errors with hypothesis testing (false conclusions being made), since they are based on the concept of probability: Type 1 error (alpha error): Rejection of the null hypothesis when it is true (i.e. reporting an important difference, when in fact at the level of population there is non. The difference observed in the sample was in fact a chance finding. It is presented by alpha, which is the level of significance, often the 5% and to a lesser extent the 0.1% (α=0.05, and 0.001) levels are chosen.

Alpha error It is the level of chance tolerated by the researcher as an alternative explanation for the results of the sample. It is of small magnitude (5% or 1/1000), but it needs to be remembered and taken into account when the stakes are high, like for example in case of dangerous disease or complications. (Thalidomide disaster is an example).

Beta Error Type I1 error (Beta error): Accepting the null hypothesis when it is false. i.e. There is a real effect or difference, but the researcher is unable to accept the alternative hypothesis. It denotes failure to detect a real effect or difference. The power of study to detect an effect is measured by “one minus Beta error”. This error is not evaluated by the type of statistical methods tough in biostatistics module. It needs more advanced statistical tools. Usually studied in relation to sample size estimation. A larger sample size will result in lower type-II error and higher study power.

Type I and Type II Errors True State of Nature The null hypothesis is true The null hypothesis is false Type I error (rejecting a true null hypothesis)  We decide to reject the null hypothesis Correct decision Decision Type II error (rejecting a false null hypothesis)  page 376 of text We fail to reject the null hypothesis Correct decision

P-value It is the smallest value of α (alpha) for which the Ho can be rejected. It gives a more precise statement about probability of rejection of Ho when it is true than the alpha level, so instead of saying the test statistic is significant or not , we will mention the exact probability of rejecting the Ho when it is true. It measures the role in chance in finding an effect or difference in the sample studied.

Steps in conducting hypothesis testing Hypothesis testing can be presented as 8 steps process: 1. Data : The nature of the data whether it consists of counts, or measurement will determine the test statistic to be used (z test, t-test, Chi-square test, ANOVA). 2. State the 2 Statistical Hypotheses : Null Hypothesis (Ho) and the alternative hypothesis( HA). If we accept the Ho we will say that the data to be tested does not provide sufficient evidence, based on the current sample to cause rejection. If Ho is rejected we say that the data are not compatible with Ho and support the alternative hypothesis (HA).

Steps in conducting hypothesis testing 3. Identify the “Test Statistic” to be used: Use the data of the sample to reach to a decision to reject or to accept the null hypothesis. The general formula for a test statistic is: Statistic - Hypothesized parameter Test Statistic = ---------------------------------------------- Standard Error (SE) of the statistic 4. Identify the distribution of test statistic: (refer to step 1) 5. Identify “Decision Rule” from statistical tables: It will tell us to reject the null hypothesis if the test statistic falls in the rejection area, and to accept it if it falls in the acceptance region

Decision Rule The critical values (tabulated value) that discriminate between acceptance and rejection regions depending on alpha level of significance. If the value of the test statistic falls in the rejection region area, it is considered statistically significant. If it falls in the acceptance area it is considered not statistically significant Whenever we reject a null hypothesis , there is always a possibility of type 1 error (rejection of Ho when it is true). This is why we should decrease this error to the least possible.

Critical value The value of the test statistic that separate the rejection region from the acceptance region Acceptance region: A set of values of the test statistic leading to acceptance of the null hypothesis (values of the test statistic not included in the critical region) Rejection region: A set of values of the test statistic leading to rejection of the null hypothesis

Steps-continued 6. Computing Test Statistic: 7. Statistical decision: It consists of rejecting or accepting (not rejecting) the Ho . It is rejected if the computed value of the test statistic falls in the rejection area (its absolute value ≥ absolute value of decision rule), and it is not rejected if computed value of the test statistic falls in the acceptance region (its absolute value < absolute value of decision rule). 8. Conclusion: If Ho is rejected , we conclude that HA is true, while If Ho is not rejected we conclude that HA may be true.

Two sided test : If the rejection area is divided into the two tails the test is called two- sided test. (this is the usual condition) One sided test: If the rejection region is only in one tail it is called one-sided test. The decision will depend on the nature of the research question being asked by the researcher. Note: If we can reject the Ho on a two sided test (statistically significant finding) it will also be significant on a one sided test. The reverse is not true.

Computing test statistic 1. Single population mean , known population variance _ X - µ Z=------------- σ /√ n 2. Single population mean with unknown population variance _ t =------------- S /√ n

Computing test statistic-continued 3. Difference between two populations mean with known variances _ _ (X1 –X2) – (µ1-µ2) Z=------------------------------- √ σ21 /n1 + σ22 /n2 4. Difference between two populations mean with unknown and unequal variances t =------------------------------- √ s21 /n1 + s22 /n2

Computing test statistic-continued 5. Difference between two populations mean with unknown but assumed equal variances _ _ (X1 –X2) – (µ1-µ2) t =------------------------------- Sp√ 1 /n1 + 1 /n2 6. Mean difference (paired t-test) _ d -µd t =------------- Sd /√ n

7. Single population proportion P – P Z = ------------- √P(1-P)/n 8 7. Single population proportion P – P Z = ------------- √P(1-P)/n 8. Difference between two population proportions (P1-P2) –(P1-P2) Z=----------------------------------------- √P1(1-P1)/n1 + P2(1-P2)/n2

Example-1 A certain breed of rats shows a mean weight gain of 65 gm, during the first 3 months of life. 16 of these rats were fed a new diet from birth until age of 3 months. The mean was 60.75 gm. If the population variance is 10 gm , is there a reason to believe at the 5% level of significance that the new diet causes a change in the average amount of weight gained

Solution Ho : µ =65 HA: µ ≠ 65 Z 1-α/2=1-(0.05/2)=0.975 α=0.05 Z=1.96 (Decision rule or critical value) _ x -µ 60.75-65 Z=---------- = ----------- = -5.38 σ /√n √10/ √16 Sine the calculated value falls in the rejection region (its absolute value ≥ absolute value of decision rule or 5.38 ≥1.96), we reject the Ho, and accept the HA. The new diet causes a statistically significant change in the average amount of weight gained.

Example-2 In the previous example , if the population variance is unknown, and the sample S (standard deviation is 3.84 t 1- α/2 = 2.13 df =n-1 _ X -µ 60.75-65 t =-------------=------------= - 4.13 s /√n 3.84/ √16 Sine the calculated values falls in the rejection region (its absolute value ≥ absolute value of decision rule or 4.13 ≥2.13), we reject the Ho, and accept the HA

Example-3 In a study two types of dental cements were used to hold a crown on tooth cast. The amount of force in foot pounds required to pull each cemented crown from the cast was reported : _ X n σ ----------------------------------------------------------- Cement 1: 45 50 4.1 Cement 2: 42 50 3.4 Test the hypothesis that µ1= µ2 at α=0.05. 2. If σ1 and σ2 are unknown but assumed to be equal test the same hypothesis if : S1=6.2 S2= 5.2. 3. If σ1 and σ2 are unknown and unequal test the same hypothesis

Example-4 In a dental clinic it is hypothesized that 90% of all 4-years old children give no evidence of dental caries. In a study of 100 children 82 gave no such evidence , would you accept the quoted value of the 90% ? Use α=0.05

Example-4 Two communities were sampled to learn about their attitudes towards “organ donation after death” prior to campaigns being launched. The results showing favorable attitude towards the concept of after death organ donation were : n1= 110 n2= 75 ˜ ˜ P1=0.52 P2=0.55 Does the two communities have equal proportions of favorable attitude.