Learning objectives • • • • What is hypothesis? Types of hypothesis Normal distribution curve Hypothesis testing Level of significance Types of errors p value One & two tail tests Degree of freedom Data analysis What is a Hypothesis? • An educated guess • A tentative point of view • A proposition not yet tested • A preliminary explanation • A preliminary Postulate • A hypothesis is a claim (assumption) about a population parameter Various authors • “A hypothesis is a conjectural statement of the relation between two or more variables”. (Kerlinger, 1956) • “Hypothesis are single tentative guesses, good hunches – assumed for use in devising theory or planning experiments intended to be given a direct experimental test when possible”. (Eric Rogers, 1966) • “Hypothesis is a formal statement that presents the expected relationship between an independent and dependent variable.”(Creswell, 1994) • A hypothesis is a logical supposition, a reasonable guess, an educated conjecture. It provides a tentative explanation for a phenomenon under investigation." (Leedy and Ormrod, 2001). A Hypothesis : • • • • • • • • must make a prediction must identify at least two variables should have an elucidating power should strive to furnish an acceptable explanation or accounting of a fact must be falsifiable meaning hypotheses must be capable of being refuted based on the results of the study must be formulated in simple, understandable terms should correspond with existing knowledge In general, a hypothesis needs to be unambiguous, specific, quantifiable, testable and generalizable. Characteristics of a Testable Hypothesis 1. A Hypothesis must be conceptually clear - concepts should be clearly defined - the definitions should be commonly accepted - the definitions should be easily communicable 2. The hypothesis should have empirical reference - Variables in the hypothesis should be empirical realities - If they are not it would not be possible to make the observation and ultimately the test 3. The Hypothesis must be specific - Place, situation and operation Characteristics of a Testable Hypothesis 4. A hypothesis should be related to available techniques of research - Either the techniques are already available or - The researcher should be in a position to develop suitable techniques 5. The hypothesis should be related to a body of theory - Hypothesis has to be supported by theoretical argumentation - It should depend on the existing body of knowledge In this way the study could benefit from the existing knowledge and later on through testing the hypothesis could contribute to the reservoir of knowledge Categorizing Hypotheses Can be categorized in different ways 1. Based on their formulation • Null Hypotheses and Alternate Hypotheses 2. Based on direction • Directional and Non-directional Hypothesis 3. Based on their derivation • Inductive and Deductive Hypotheses The Null Hypothesis, H0 • States the claim or assertion to be tested • Is always about a population parameter, not about a sample statistic • Begin with the assumption that the null hypothesis is true – Similar to the notion of innocent until proven guilty • Refers to the status quo • Always contains “=” , “≤” or “” sign • May or may not be rejected • It states that independent variable has no effect and there will be no difference b/w the two groups. The Alternative Hypothesis, H1 • • • • • Is the opposite of the null hypothesis Challenges the status quo Never contains the “=” , “≤” or “” sign May or may not be proven Is generally the hypothesis that the researcher is trying to prove • It states that independent variable has an effect and there will be a difference b/w the two groups. Categorizing Hypotheses (Cont…) 2. Directional Hypothesis and Non-directional Hypothesis • Simply based on the wording of the hypothesis we can tell the difference between directional and non-directional – If the hypothesis simply predicts that there will be a difference between the two groups, then it is a non-directional hypothesis. It is nondirectional because it predicts that there will be a difference but does not specify how the groups will differ. – If, however, the hypothesis uses so-called comparison terms, such as “greater,”“less,”“better,” or “worse,” then it is a directional hypothesis. It is directional because it predicts that there will be a difference between the two groups and it specifies how the two groups will differ 3. Inductive and Deductive Hypotheses(Theory Building and Theory Testing) • classified in terms of how they were derived: - Inductive hypothesis - a generalization based on observation Observation Pattern Hypothesis Theory - Deductive hypothesis - derived from theory Theory Hypothesis Observation Confirmation Normal Distribution Curve •A normal distribution curve is symmetrical, bellshaped curve defined by the mean and standard deviation of a data set. •The normal curve is a probability distribution with a total area under the curve of 1. •The mean of the data in a standard normal distribution is 0 and the standard deviation is 1. •A standard normal distribution is the set of all z-scores One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99.7 percent of the data. 6 Steps in Hypothesis Testing 1. State the null hypothesis, H0 and the alternative hypothesis, H1 2. Choose the level of significance, , and the sample size, n 3. Determine the appropriate test statistic (two-tail, one-tail, and Z or t distribution) and sampling distribution 4. Determine the critical values(mainly three criteria, (i) significance level,(ii) degree of freedom,(iii) One or two tailed test,that divide the rejection and non rejection regions 5. Collect data and compute the value of the test statistic 6. Make the statistical decision and state the managerial conclusion. If the test statistic falls into the non rejection region, do not reject the null hypothesis H0. If the test statistic falls into the rejection region, reject the null hypothesis. ExpresCshatph9e-1m5anagerial conclusion in the context of the problem Steps in Hypothesis Testing Problem Definition Clearly state the null and alternate hypotheses. Choose the relevant test and the appropriate probability distribution Determine the significance level Choose the critical value Determine the degrees of freedom Compute relevant test statistic Compare test statistic and critical value Decide if one-or two-tailed test Does the test statistic fall in the critical region? Yes Reject null No Do not reject null Level of Significance, • Defines the unlikely values of the sample statistic if the null hypothesis is true • Indicates the percentage of sample means that is outside the cut-off limits (critical value) • It is the max. value of probablity of rejecting null hypothesis when it is true. – Defines rejection region of the sampling distribution • Is designated by , (level of significance) – Typical values are 0.01, 0.05, or 0.10 • Is selected by the researcher at the beginning • Provides the critical value(s) of the test Level of Significance and the Rejection Region Level of significance = //2 H0: μ = 3 H1: μ ≠ 3 Two-tail test Represents critical value //2 Rejection region is shaded 0 H0: μ ≤ 3 H1: μ > 3Upper-tail test 0 H 0: μ ≥ 3 H1: μ < 3Lower-tail test 0 Errors in Making Decisions • Type I Error – Reject a true null hypothesis – Considered a serious type of error • The probability of Type I Error is • Called level of significance of the test • Set by the researcher in advance • Type II Error – Fail to reject a false null hypothesis • The probability of Type II Error is β Testing of hypotheses Type I and Type II Errors No study is perfect, there is always the chance for error Decision Accept H0 / reject HA Reject H0 /accept HA H0 true / HA false H0 false / HA true Type II error () OK p=1- Type I error () p= p= OK p=1- - level of 1- - power of significance the test Testing of hypotheses Type I and Type II Errors α =0.05 there is only 5 chance in 100 that the result termed "significant" could occur by chance alone The probability of making a Type I (α) can be decreased by altering the level of significance. it will be more difficult to find a significant result the power of the test will be decreased the risk of a Type II error will be increased Type I & II Error Relationship Type I and Type II errors cannot happen at the same time Type I error can only occur if H0 is true Type II error can only occur if H0 is false If Type I error probability ( ) Type II error probability ( β ) , then Factors affecting type II error All else equal: – β when the difference between hypothesized parameter and its true value – β when – β when σ – β when n Testing of hypotheses Type I and Type II Errors The probability of making a Type II () can be decreased by increasing the level of significance. it will increase the chance of a Type I error To which type of error you are willing to risk ? Degree of Freedom • The number or bits of "free" or unconstrained data used in calculating a sample statistic or test statistic • It refers to the scores in a distribution that are free to change without changing the mean of distribution. • A sample mean (X) has `n' degree of freedom • A sample variance (s2) has (n-1) degrees of freedom • This no. is used to determine power ,because the more subjects the greater the power One-Tail Test • In many cases, the alternative hypothesis focuses on a particular direction • Determines whether a particular population parameter is larger or smaller than some predefined value • Uses one critical value of test statistic H0: μ ≥ 3 H1 : μ < 3 H0: μ ≤ 3 H1: μ > 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 Two tailed test • Two-tailed Test the • Determines likelihood that a population parameter is within certain upper and lower bounds • May use one or two critical values Confidence interval and significance test Null hypothesis is accepted A value for null hypothesis within the 95% CI p-value > 0.05 Null hypothesis is rejected A value for null hypothesis outside of 95% CI p-value < 0.05 p-Value Approach to Testing • p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H0 is true • Also called observed level of significance • Smallest value of for which H0 can be rejected p-Value Approach to Testing (continued) • Convert Sample Statistic (e.g., X ) to Test Statistic (e.g., Z statistic ) • Obtain the p-value from a table or computer • Compare the p-value with – If p-value < , reject H0 – If p-value , do not reject H0 The First Question After examining your data, ask: does what you're testing seem to be a question of relatedness or a question of difference? •If relatedness (between your control and your experimental samples or between you dependent and independent variable), We will be using tests for correlation (positive or negative) or regression. •If difference (your control differs from your experimental), we will be testing for independence between distributions, means or variances. Different tests will be employed if your data show parametric or non-parametric properties. Parametric or Non-parametric Parametric tests: to estimate at least one population parameter from sample statistics and are restricted to data that: 1) show a normal distribution 2) are independent of one another 3) are on the same continuous scale of measurement 4) require certain assumptions about the parameters of the population such as knowing μ and Non-parametric tests : are used on data that: 1) show an other-than normal distribution 2) are dependent or conditional on one another 3) in general, do not have a continuous scale of measurement 4) does not require assumptions about the parameters of population such as knowing μ and are not needed the Parametric and nonparametric tests of significance N o n p a r a m e t r i c tests One group Two unrelated groups T w o related groups K-unrelated groups K-related groups Nominal data Chi square goodness o f fit Chi square Ordinal data Wilcoxon signed rank test Wilcoxon rank s u m t e s t , MannWhitney t e s t McNemar’s t e s t W i l c o x o n signed rank test Chi square Kruskal -Wallis test one way analysis of variance Friedman matched samples Pa ra metri c tests Ordinal, interval, ratio data O n e g r o u p t-test Student’s t-test Paired Student’s t-test A N O V A A N O V A with repeated measurements Types of Parametric tests 1. Large sample tests Z-test 2. Small sample tests t-test * Independent/ unpaired t-test * Paired t-test ANOVA (Analysis of variance) * One way ANOVA * Two way ANOVA 41 Z test: • It is used to test the null hypothesis for a single sample when the population variance is known. • A z-test is used for testing the mean of a population versus a standard, or comparing the means of two populations, with large (n ≥ 30) samples whether you know the population standard deviation or not • It is used to judge the significance of several statistical measures ,particularly mean. • It compares a sample mean with the sampling distribution, i.e the sample is part of the sampling distribution • It is also used for testing the proportion of some characteristic versus a standard proportion, or comparing the proportions of two populations. Ex. Comparing the average engineering salaries of men versus women. Ex. Comparing the fraction defectives from two production lines. 44 Formula in Computing the Test Statistic Using Z Test (Two Sample Mean Test) • when the given means are sample means. 𝒙₁ − 𝒙₂ 𝒛= 𝒔₁² 𝒔₂² 𝒏₁ + 𝒏₂ • when the given means are population means. 𝒛= 𝝁₁ − 𝝁₂ 𝝈₁² 𝝈₂² + 𝒏₁ 𝒏₂ 𝒙₁ = mean of the 1st sample 𝒙₂ = mean of the 2nd sample 𝝁₁ = mean of the 1st population 𝝁₂ = mean of the 2nd population 𝐬₁ = standard deviation of the 1st sample 𝐬₂ = standard deviation of the 2nd sample 𝝈₁ = standard deviation of the 1st population 𝝈₂ = standard deviation of the 2nd population 𝒏₁ = size of the 1st sample or population 𝒏₂ = size of the 2nd sample or population One tailed Z test: • A directional test in which a prediction is made that the population represented by sample is either below or above the general population Ha: μ 0 < μ 1 or Ha: μ 0 > μ 1 Two tailed Z test: • A non directional test in which a prediction is made that the population represented by sample will differ from the general population,but thre direction of the difference is not predicted Ha: μ 0 ≠ μ 1 Example • Ho: Children who learn whole language approach do not statistically significantly differ from the average child in word recognition (µ = 75%, σ = 5%). In symbols: Ho: µ = 75%. H1: Children who learn whole language approach statistically significantly differ from the average child with respect to word recognition (µ = 75%, σ = 5%). In symbols: H1: µ ≠ 75%. •α = 0.05, thus the critical values (C.V.) are ± 1.96. Sample mean= 78% Population mean = 75% σ = 5% n = 50 Z= x . = 78 - .75 = 0.03 = 4.24(This is the test statistic which is a z – .05 /√50 .05 /√50 score (unit: standard deviation n We reject the null hypothesis and conclude that children who learn the whole language statistically significantly differ from the average child in word recognition, z = 4.24, p < .05. t test: Derived by W S Gosset in 1908 • It is based on t-distribution • It is the indicator of the no. of standard deviation units the sample mean is from the mean of the sampling distribution • Used to judge the significance of a smaple mean or for judging the difference b/w the means of 2 samples in case of small sample(usually < 30) when population variance is not known, • Properties of t distribution: i. It has mean 0 ii. It has variance greater than one iii. It is bell shaped symmetrical distribution about mean • Assumption for t test: i. Sample must be random, observations independent ii. Standard deviation is not known iii. Normal distribution of population Uses of t test: i. The mean of the sample ii. The difference between means or to compare two samples iii. Correlation coefficient Types of t test: a. Paired t test b. Unpaired t test Paired t test: • Consists of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test). • Ex. where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication Unpaired t test: • When two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. • Ex: 1. compare the height of girls and boys. 2. compare 2 stress reduction interventions when one group practiced mindfulness meditation while the other learned progressive muscle relaxation. One tailed t test: • A directional test in which a prediction is made that the population represented by sample is either below or above the general population Ha: μ 0 < μ 1 or Ha: μ 0 > μ 1 Two tailed t test: • A non directional test in which a prediction is made that the population represented by sample will differ from the general population, but the direction of the difference is not predicted Ha: μ 0 ≠ μ 1 ANOVA • Prof R. A fisher was the first to use the term variance and developed a theory concerning ANOVA • ANOVA (Analysis of Variance) compares the means of two or more parametric samples. • It tests the difference among different groups of data for homogenity • Basic principle of ANOVA is to test for differences among the means of the populations by examining the amount of variation within each of these samples, relative to the amount of variation b/w samples. • The statistic for ANOVA is called the F statistic, which we get from the F Test F = Estimate of population variance based on b/w sample variance Estimate of population variance based on within sample variance • If we take one factor and investigate the differences amongst it various categories we use one way ANOVA • In case we investigate 2 factors at the same time ,then we use two way ANOVA • The ANOVA test has 2 degrees of freedom: – N-I (Total number sampled – Number of Groups) – I-1 (Number of Groups – 1) • Assumption for ANOVA test: i. Normal distribution of population ii. 3 or more groups, iii. Variables are independent iv. Data is interval or ratio, v. Homogenity of variance Difference between one & two way ANOVA • An example of when a one-way ANOVA could be used is if we want to determine if there is a difference in the mean height of stalks of three different types of seeds. Since there is more than one mean, we can use a one-way ANOVA since there is only one factor that could be making the heights different. • Now, if we take these three different types of seeds, and then add the possibility that three different types of fertilizer is used, then we would want to use a two-way ANOVA. • The mean height of the stalks could be different for a combination of several reasons • The types of seed could cause the change, the types of fertilizer could cause the change, and/or there is an interaction between the type of seed and the type of fertilizer. • There are two factors here (type of seed and type of fertilizer), so, if the assumptions hold, then we can use a two-way ANOVA. Summary of parametric tests applied for different type of data Sl no Type of Group Parametric test 1. Comparison of two paired groups Paired ‘t’ test 2. Comparison of two unpaired groups Unpaired ‘t’ test 3. Comparison of three or more matched groups Two way ANOVA 4. Comparison of three or more matched groups One way ANOVA 5. Correlation between two variables Pearson correlation 60 Commonly used non parametric tests • Commonly used Non Parametric Tests are: − Chi Square test − The Sign Test − Wilcoxon Signed-Ranks Test − Mann–Whitney U or Wilcoxon rank sum test − The Kruskal Wallis or H test − Friedman ANOVA − The Spearman rank correlation test − Cochran's Q test Chi Square test • First used by Karl Pearson • Simplest & most widely used non-parametric test in statistical work. • Calculated using the formulaχ2 = ∑ ( O – E )2 E O = observed frequencies E = expected frequencies • Greater the discrepancy b/w observed & expected frequencies, greater shall be the value of χ2. • Calculated value of χ2 is compared with table value of χ2 for given degrees of freedom. Chi Square test • Application of chi-square test: – Test of association (smoking & cancer, treatment & outcome of disease, vaccination & immunity) – Test of proportions (compare frequencies of diabetics & nondiabetics in groups weighing 40-50kg, 50-60kg, 60-70kg & >70kg.) – The chi-square for goodness of fit (determine if actual numbers are similar to the expected/theoretical numbers) Chi Square test • Attack rates among vaccinated & unvaccinated children against measles : Group Result Total Attacked Not-attacked Vaccinated (observed) (a)10 (b) 90 (a+b)100 Unvaccinated (observed) (c) 26 (d) 74 (c+d) 100 Total (a+c) 36 (b+d) 164 200 • Prove protective value of vaccination by χ2 test at 5% level of significance Chi Square test Group Total Result Attacked Not-attacked Vaccinated (Expected) 18 82 100 Unvaccinated (Expected) 18 82 100 Total 36 164 200 Chi Square test χ2 value = ∑ (O-E)2/E (10-18)2 + (90-82)2 + (26-18)2 +(74-82)2 18 82 18 82 64 + 64 + 64 + 64 18 82 18 82 =8.67 calculated value (8.67) > 3.84 (expected value corresponding to P=0.05) Direct formula = (ad-bc)2 * N (a+b)(c+d)(a+c)(b+d) Null hypothesis is rejected. Vaccination is protective. • Yates’ correction: applies when we have two categories (one degree of • • • freedom) Used when sample size is ≥ 40, and expected frequency of <5 in one cell Subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table χ2 = ∑ [O- E-0.5]2 E The Chi-Square Test for Goodness-of-Fit (cont.) • The null hypothesis specifies the proportion of the population that should be in each category. • The proportions from the null hypothesis are used to compute expected frequencies that describe how the sample would appear if it were in perfect agreement with the null hypothesis. The Chi-Square Test for Independence • The second chi-square test, the chi-square test for independence, can be used and interpreted in two different ways: 1. Testing hypotheses about the relationship between two variables in a population, or 2. Testing hypotheses about differences between proportions for two or more populations. Sign Test • • • • • • Used for paired data, can be ordinal or continuous Simple and easy to interpret Makes no assumptions about distribution of the data Not very powerful To evaluate H0 we only need to know the signs of the differences If half the differences are positive and half are negative, then the median = 0 (H0 is true). • If the signs are more unbalanced, then that is evidence against H0. Sign Test – Children in an orthodontia study were asked to rate how they felt about their teeth on a 5 point scale. – Survey administered before and after treatment. How do you feel about your teeth? 1. Wish I could change them 2. Don’t like, but can put up with them 3. No particular feelings one way or the other 4. I am satisfied with them 5. Consider myself fortunate in this area child Rating before Rating after 1 1 5 2 1 4 3 3 1 4 2 3 5 4 4 6 1 4 7 3 5 8 1 5 9 1 4 10 4 4 11 1 1 12 1 4 13 1 4 14 2 4 15 1 4 16 2 5 17 1 4 18 1 5 19 4 4 20 3 5 • Use the sign test to evaluate whether these data provide evidence that orthodontic treatment improves children’s image of their teeth. child Rating before Rating after change 1 1 5 4 2 1 4 3 3 3 1 -2 4 2 3 1 5 4 4 0 6 1 4 3 difference between the two 7 3 5 2 ratings 8 1 5 4 9 1 4 3 10 4 4 0 11 1 1 0 12 1 4 3 13 1 4 3 14 2 4 2 15 1 4 3 16 2 5 3 17 1 4 3 18 1 5 4 19 4 4 0 20 3 5 2 • First, for each child, compute the child Rating before Rating after change sign 1 1 5 4 + 2 1 4 3 + 3 3 1 -2 - 4 2 3 1 + 5 4 4 0 0 6 1 4 3 + 7 3 5 2 + 8 1 5 4 + 9 1 4 3 + 10 4 4 0 0 11 1 1 0 0 12 1 4 3 + 13 1 4 3 + 14 2 4 2 + 15 1 4 3 + 16 2 5 3 + 17 1 4 3 + 18 1 5 4 + 19 4 4 0 0 20 3 5 2 + • The sign test looks at the signs of the differences – 15 children felt better about their teeth (+ difference in ratings) – 1 child felt worse (- diff.) – 4 children felt the same (difference = 0) • If H0 were true we’d expect an equal number of positive and negative differences. (P value from table 0.004) Wilcoxon signed-rank test • Nonparametric equivalent of the paired t-test. • Similar to sign test, but take into consideration the magnitude of difference among the pairs of values. (Sign test only considers the direction of difference but not the magnitude of differences.) For eg • The 14 difference scores in BP among hypertensive patients after giving drug A were: -20, -8, -14, -12, -26, +6, -18, -10, -12, -10, -8, +4, +2, -18 • The statistic T is found by calculating the sum of the positive ranks, and the sum of the negative ranks. • The smaller of the two values is considered. 74 Wilcoxon signed-rank test • • • • • • • • • • • • • • Score +2 +4 +6 -8 -8 -10 -10 -12 -14 -16 -18 -18 -20 -26 Rank 1 2 3 4.5 4.5 6.5 6.5 8 9 10 11.5 11.5 13 14 Sum of positive ranks = 6 Sum of negative ranks = 99 T= 6 For N = 14, and α = .05, the critical value of T = 21. If T is equal to or less than T critical, then null hypothesis is rejected i.e., drug A decreasesthe BP among hypertensive patients. Mann-Whitney U test • Mann-Whitney U – similar to Wilcoxon signed-ranks test except that the samples are independent and not paired. • Null hypothesis: the population means are the same for the two groups. • Rank the combined data values for the two groups. Then find the average rank in each group. • Then the U value is calculated using formula • U= N1*N2+ Nx(Nx+1) _ Rx (where Rx is larger rank total) 2 • To be statistically significant, obtained U has to be equal to or LESS than this critical value. Example • 10 dieters following Atkin’s diet vs. 10 dieters following Jenny Craig diet Hypothetical RESULTS: • Atkin’s group loses an average of 34.5 lbs. • J. Craig group loses an average of 18.5 lbs. • Conclusion: Atkin’s is better? • When individual data is seen • Atkin’s, change in weight (lbs): +4, +3, 0, -3, -4, -5, -11, -14, -15, -300 •J. Craig, change in weight (lbs) -8, -10, -12, -16, -18, -20, -21, -24, -26, -30 • RANK the values, 1 being the least weight loss and 20 being the most weight loss. • Atkin’s – +4, +3, 0, -3, -4, -5, -11, -14, -15, -300 – 1, 2, 3, 4, 5, 6, 9, 11, 12, 20 • J. Craig − -8, -10, -12, -16, -18, -20, -21, -24, -26, -30 − 7, 8, 10, 13, 14, 15, 16, 17, 18, 19 • Sum of Atkin’s ranks: 1+ 2 + 3 + 4 + 5 + 6 + 9 + 11+ 12 + 20=73 • Sum of Jenny Craig’s ranks: 7 + 8 +10+ 13+ 14+ 15+16+ 17+ 18+19=137 • Jenny Craig clearly ranked higher. • Calculated U value (18) < table value (27), Null hypothesis is rejected. Kruskal-Wallis One-way ANOVA • It’s more powerful than Chi-square test. • It is computed exactly like the Mann-Whitney test, except that there are more groups (>2 groups). • Applied on independent samples with the same shape (but not necessarily normal). Friedman ANOVA • Friedman ANOVA: When either a matched-subjects or repeatedmeasure design is used and the hypothesis of a difference among three or more (k) treatments is to be tested, the Friedman ANOVA by ranks test can be used. Spearman rank-order correlation • Use to assess the relationship between two ordinal variables or two skewed continuous variables. • Nonparametric equivalent of the Pearson correlation. • It is a relative measure which varies from -1 (perfect negative relationship) to +1 (perfect positive relationship).