7 Key excerpts on "Hypothesis Test for Correlation"

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  Mathematics for Enzyme Reaction Kinetics and Reactor Performance

  • F. Xavier Malcata(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  19 Statistical Hypothesis Testing

  A statistical hypothesis is a hypothesis that is testable, on the basis of observing a process modeled by a set of random variables that follow a probability distribution known a priori. Hypothesis testing (or confirmatory data analysis) is a method of statistical inference that resorts to tests of significance to determine the probability that a statement is true, and at what likelihood such a statement may be accepted as true. Usually, two datasets are compared, or data obtained from sampling is compared against synthetic data produced via an idealized model; a working hypothesis is then put forward for the statistical relationship between the data, versus a complementary hypothesis.

  Therefore, the basic process of hypothesis testing consists of four sequential steps: (i) formulate the null hypothesis, H0 (commonly stated as if the observations are the result of pure chance), and the alternative hypothesis, H1 (commonly stated as if the observations unfold an actual effect, along with an unavoidable component of variation by chance); (ii) identify an appropriate test statistic that can be used to assess the truth of the null hypothesis (dependent on the nature of the data and of the test); (iii) compute the P‐value, or associated probability that a test statistic at least as significant as the one determined from the sample data would be obtained, assuming that the null hypothesis held true (the smaller said probability, the stronger the evidence against the null hypothesis); and (iv) compare the P‐value with an acceptable significance level, α (or threshold of probability, often 1% or 5%) – if P ≤ α, then the observed effect is statistically significant, so the null hypothesis is ruled out, and the alternative hypothesis is concomitantly accepted as valid.

  The probability of rejecting the null hypothesis (H0 , expressed as =) is a typical function of five factors – whether the test is two‐tailed (i.e. H1 is expressed as ≠) or one‐tailed (i.e. H1 is expressed as either < or >), the value of α, the intrinsic variance of the data, the amount of deviation from H0 , and the size of the sample. This rationale is illustrated in Fig. 19.1 . Decision on whether or not to accept the null hypothesis is based on the actual value taken by the test statistic – the distribution of which is specified on the assumption that H0 holds true, and typically follows one of the continuous distributions discussed so far; if said value is very unlikely, then H0 should be rejected and H1 concomitantly accepted. In order to take this decision on a quantitative basis, a probability value α must be chosen a priori – below which H0 becomes sufficiently unlikely to be (safely) rejected; the portion(s) of the area below the probability density function curve accounting for α will be continuous whenever a unilateral test is at stake, or else splitted (equally) in half in the case of a bilateral test – with which is which being determined by the nature of H1

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  Correlation and Regression

  Applications for Industrial Organizational Psychology and Management

  • Philip Bobko(Author)
  • 2001(Publication Date)
  • SAGE Publications, Inc

    (Publisher)

  Chapter I ). In fact, testing Pearson product moment correlations for significance is relatively straightforward, although the procedures depend upon the type of hypothesis being tested.

  TESTING A SINGLE CORRELATION AGAINST ZERO

  The hypothesis H 0 : ρ = 0, where ρ is the underlying population correlation, is by far the most frequently tested correlational hypothesis. In fact, many researchers run around asking each other, “Was your r significant?” This is researcher shorthand for, “Was your sample value of r significantly different from the hypothesized value of ρ = 0?”or “Assuming the population correlation was 0, was your sample value of r so far away from 0 that it was not likely to have occurred by chance?”

  Notationally, let ρ be the value of the population correlation, let r be our old friend the sample correlation, and let n be the sample size. Then, it can be shown that the value has a t -distribution with (n − 2) degrees of freedom. Formally, we have

  Note that the value of r is tested by computing a fairly straightforward (but nonlinear) function of r and comparing the result to well-known critical values of Student’s t -distribution.2 For example, suppose the predictive validity of the Scholastic Aptitude Test (SAT) is being questioned. Assume that a researcher uses a simple random sample of n = 122 and obtains a measure of success in college (Y ) as well as previous SAT scores (X ) for each individual. Suppose the resulting correlation is r = .27. Then, Equation III.A gives

  The obtained sample value of t = 3.07 is larger than the two-sided, p = .05 critical t -value of 1.980 (see Appendix, Table A.1 ). Therefore, this correlation is “significant.” More correctly stated, we have rejected the null hypothesis (that the underlying value of ρ is 0) in favor of the alternative hypothesis that ρ is nonzero (but you’ll usually just hear the cry, “My r is significant!”). [By the way, the same value of 3.07 is statistically significant at the p

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  Research Methods for Public Administrators

  • Gary Rassel, Suzanne Leland, Zachary Mohr, Elizabethann O'Sullivan(Authors)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Tests of statistical significance allow a researcher to determine the probability that variables related in a random sample are not related in the population. A test of statistical significance cannot indicate that a relationship is strong or important; it may not even indicate if the direction is as hypothesized. Nor does a finding of statistical significance imply that other parts of the research were carried out correctly. The test only makes a statistical statement about the nature of a relationship.

  To carry out a test of statistical significance, a researcher

  1. states the null and the research hypotheses
  2. selects an alpha level
  3. selects and computes a test statistic
  4. makes a decision

  Figure 12.4 illustrates these steps.

  Figure 12.4

  Testing Statistical Significance of a Relationship

  The null hypothesis postulates that an independent and a dependent variable are not related. For some statistical tests, the null hypothesis may state that the relationship goes in a certain direction or does not exceed a specific value.

  In hypothesis testing, a researcher runs the risk of making two errors. First, he may reject a null hypothesis that is actually true. The researcher, then, has “accepted” an untrue research hypothesis. This error is called a Type I error. Second, the researcher may accept a null hypothesis that is untrue. The researcher has failed to accept a true research hypothesis; that is called a Type II error.

  In practice, researchers are more likely to make a judgment about a hypothesis based on the associated probability, the sample size, the magnitude of the effect, and the practical consequences of their decisions than they are to decide on a specific alpha level set a priori. The associated probability is the probability of a specific Χ2 or t

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  Econometrics

  • K. Nirmal Ravi Kumar(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  O ) is that:

  HO : ρ = 0 (there is no relationship between the two variables X and Y, when all population values are observed) and an alternative hypothesis (HA ) is framed against this as:

  HA : Alternative hypothesis could be any one of the following three forms viz., (ρ≠0) or (ρ<0) or (ρ>0). That is, if the researcher has no idea whether or how two variables are related, then a two tailed HA ie., HA : (ρ≠0) is formulated. If the researcher suspects, or has knowledge, that the two variables are negatively related then, HA : (ρ<0) is formulated (ie., one tailed HA ) and if the researcher predicts a positive relationship between the variables, then HA : (ρ>0) is formulated (ie., one tailed HA ).

  The test statistic for the hypothesis test is the sample or observed correlation coefficient ‘r’ obtained from a sample. The sampling distribution of ‘r’ is approximated by a ‘t’ distribution with n–2 Degrees of Freedom (df). The formula to compute ‘tcal ’ value is given by:

  t = r S E ( r )

  Equation 2.14

  where, r = Sample correlation coefficient, Standard Error (SE) of

  ‘ r ’ =

  1 −

  r 2

  n − 2

  n = number of observations. So, considering the formula of SE (‘r’), we can write:

  t = r n − 2 1 −

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  Choosing and Using Statistics

  A Biologist's Guide

  • Calvin Dytham(Author)
  • 2011(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  1 It is quite rare to find two variables that are normally distributed and therefore suitable for Pearson’s correlation. Test the data to see if they follow a normal distribution. Consider the alternatives.
• 2 The statistical significance of correlation is not a good guide to the real significance of the correlation. With large sample sizes the value of r required to achieve statistical significance (i.e. to show that there is some relationship between the two variables) is rather low. It is perhaps better to use the value of r 2 as an indicator of the real significance as this value shows the amount of variation in one variable explained by the other.

An example

A marine biologist working on Adélie penguins (Pygoscelis adeliae) has measured the sizes of birds forming pairs. The measure used is the length of a bone in the leg which is known, from previous studies, to be a good indication of size. It is measured to the nearest 0.1 mm. The null hypothesis is that male size is not correlated with female size. Unfortunately data could only be collected from six pairs.

Pair	Female	Male
1	17.1	16.5
2	18.5	17.4
3	19.7	17.3
4	16.2	16.8
5	21.3	19.5
6	19.6	18.3

It is assumed that both variables are normally distributed. This data set is a little small to test this although a larger sample of the population might be more suitable (see ‘Do frequency distributions differ?’ in the previous chapter, page 72). In this case the null hypothesis is rejected as there is a significant positive correlation between male and female size indicating that there is positive, assortative mating in this species. The value of r is 0.88 and r 2 is 0.77 indicating that 77% of the variation in the size of one sex is explained by the size of the other.

SPSS This is one of the easiest tests to carry out in SPSS. Input the data in two columns and add appropriate column labels. The cases (pairs in the example) do not require a separate column. From the ‘Analyze’ menu, select ‘Correlate’ then ‘Bivariate…’. In the dialogue box move the names of the two variables into the ‘Variables:’ box and make sure that the ‘Pearson’ option and ‘Two-tailed’ are checked. Click ‘OK’.