Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations and SLR in descriptive manner with no assumptions. Now consider using correlations and SLR in inferential manner: Making inferences about population from which sample was drawn. Hypothesis tests. Confidence intervals. Such analyses and inferences require use of sampling distributions and knowledge of their properties. To define properties of sampling distributions we must make distributional assumptions. Assumptions of Fixed Linear Regression Model: is fixed; levels are selected. Y is randomly sampled at each level of.
2 Regression residuals are normal, and variance of residuals is same at each level of (homogeneity of variance). No assumption made about distribution of or about full distribution of Y. Alternative assumption when is not fixed: Joint distribution of and Y in population is bivariate normal. This means that is normal, Y is normal, and there can be no nonlinear relationship. Later we will study these assumptions further, along with consequences of violating them. For now: Results of inferential statistical methods are fairly robust against violations. Inferences in SLR: Recall correlation and SLR in sample of n observations: Pearson correlation: r Y SLR: Yˆ +
3 Sample statistics of interest: r Y,, Corresponding population parameters: Population correlation coefficient: ρ Y, or just ρ Population regression intercept: Population regression coefficient: Wish to make inferences about population parameters based on sample statistics. Two types of inferences: Confidence intervals, hypothesis tests. Consider general approach: Population parameter: θ Sample statistic (point estimate): θˆ Suppose sampling distribution of θˆ is approximately normal when n is large, with mean θ and standard error. θˆ To construct a confidence interval for θ : Choose α for CI of width 1(1-α)%. (e.g., for 95% CI, α.5)
4 Given degrees of freedom (df), determine value of t that cuts off area of α/2 in each tail of t- distribution. Call this value t α/2. Then CI is defined as: ˆ θ t θ ˆ θ + t α / 2 ˆ θ α / 2 ˆ θ For intervals computed in this fashion, the specified percentage of such intervals will include the value of the parameter, under the stated assumptions. To conduct a hypothesis test about the parameter θ : State the null hypothesis: H : θ θ (often H : θ ) State the alternative hypothesis: Non-directional (implies two-tailed test): H : θ θ Directional (implies one-tailed test): H : θ < θ 1 or H : θ > θ 1
Under the null hypothesis and the stated assumptions, the sampling distribution of θˆ will be approximately normal when n is large, with mean θ, and standard error. θˆ To test null hypothesis, obtain test statistic: ˆ θ θ t ˆ θ Choose α to represent desired Type I error rate and determine critical value of t that cuts off appropriate area in tail(s) of distribution (α for one-tailed test, and α/2 for two-tailed test). Call this value t c. Compare observed value of test statistic, t, to critical value, t c. If observed value is more extreme than critical value, reject H. This means that H is highly unlikely given observed data. If not, then fail to reject H, meaning that H is not highly unlikely given observed data. 5
6 Confidence intervals vs. hypothesis tests: CI implies result of hypothesis test; if CI does not contain θ, then H will be rejected. Width of CI provides information about precision of point estimate, which is not provided by hypothesis test. CIs provide all information that hypothesis tests provide, and more. Confidence intervals and hypothesis tests for specific parameters in correlation and SLR analyses: We apply the general framework just described in order to make inferences about parameters in correlation and regression. Inferences about regression coefficients in SLR: Population parameter: Sample statistic: Standard error: sd sd Y 2 (1 r ) ( n 2) Confidence interval: tα / 2 + tα / 2
7 Hypothesis test: Specify null and alternative hypothesis about Most common and interesting:. H : H : 1 This H implies no effect of Y on, or no linear association. Compute test statistic: t Determine t c, where df (n-2), and make decision about H. Interpret with respect to H : likelihood of whether relationship in population is zero. Inferences about regression intercept in SLR Population parameter: Sample statistic: 2 Y ( n 1) sd Standard error: Y Yˆ + 2 1 n
8 Confidence interval: tα / 2 + tα / 2 Hypothesis test: Specify null and alternative hypothesis about. Most common: H H : 1 : This H implies that in population, when, then predicted value of Y is zero. Compute test statistic: t Determine t c, where df (n-2), and make decision about H. Interpret with respect to H : likelihood of whether intercept in population is zero. Inferences about Pearson correlation coefficient Population parameter: ρ Sample statistic: r The general approach described above must be modified slightly because the sampling distribution
9 of r is not normal, but is skewed, with skewness increasing as ρ increases. To overcome this problem, Fisher developed a transformation of r into Fisher s z which more closely follows a normal distribution. This is known as Fisher s r-to-z transformation. To reduce confusion, this z will be designaged z. z' 1 [ln(1 2 + r) ln(1 r)] Values of z corresponding to any given value of r are routinely provided in tables (see Table in text). The standard error of Fisher s z is given by: z ' 1 ( n 3) Confidence interval for ρ : First establish a CI for Fisher s z according to: z' t α / 2z' ρ z' + t α / 2 z' Then transform confidence limits to correlations by using Table.
1 Hypothesis test for correlation coefficient: Specify null and alternative hypotheses. H : ρ ρ H : ρ ρ 1 Convert observed and hypothesized values of correlation into Fisher s z and conduct test by conventional methods: Convert r into z s, sample value of z. Convert ρ into z, hypothesized value of z. Compute test statistic: t z ' s z z' ' Compare observed t to critical t c and make decision about H. Interpret result in terms of likelihood that population correlation is equal to hypothesized value. Note that the most common null hypothesis about ρ is H : ρ, or that there is no linear relationship between and Y. In this case, the test statistic reduces to ' zs t z'
11 Testing other hypotheses about correlation coefficients It is common to wish to test a variety of other hypotheses about correlations, or to set up other types of confidence intervals. Other common situations include: Making inferences about the difference between two (or more) independent correlations; that is, values of r Y obtained from two (or more) different samples. Making inferences about the difference between two dependent correlations; that is values of r Y and r W obtained from the same sample. Testing the null hypothesis that a set of k measured variables are all uncorrelated with each other in the population. It is possible to perform virtually any test about patterns of correlation coefficients. For a general reference and software, see: Steiger, J.H. (23). Comparing correlations. In A. Maydeu-Olivares (Ed.) Psychometrics. A festschrift to Roderick P. McDonald. Mahwah, NJ: Lawrence Erlbaum Associates, in press. (available at Steiger s website at http://www.statpower.net/)
12 Inferences about predicted scores on Y The SLR approach provides a regression equation of the form Yˆ + that can be used to produce a predicted value of Y given any value of. Any such predicted score is likely to be in error, different from the actual value of Y for the individual. We can assess the likely degree of error by establishing a confidence interval around a predicted Y, and defining our confidence that the given interval will contain the observed Y. Under regression assumptions, for any given, predicted scores are normally distributed. Their standard error can be shown to be Yˆ i Y Yˆ 1 ( i ) + n ( n 1) sd A very important feature of this expression is that the standard error will depend on the deviation of i from the mean of. In other words, predictions of Y become less precise for individuals farther from the mean of. 2 2
13 Given this standard error, a CI for a particular predicted score can then be obtained by Yˆ i t α / 2 Y Yˆ i Yi + ˆ i Y i t α / 2 These confidence intervals can be obtained from most regression software for each individual in the sample. Or they can be computed easily for new individuals for whom predictions are made. ˆ Experience with these CIs shows that they are typically quite wide, indicating that individual predictions are often subject to much error. These CIs tend to become wider as: An individual deviates further from the mean of Sample size becomes smaller The correlation between and Y becomes smaller