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LECTURE 11 Hypotheses about Correlations

LECTURE 11 Hypotheses about Correlations. EPSY 640 Texas A&M University. Hypotheses about Correlations. One sample tests for Pearson r Two sample tests for Pearson r Multisample test for Pearson r Assumptions: normality of x, y being correlated. One Sample Test for Pearson r.

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LECTURE 11 Hypotheses about Correlations

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  1. LECTURE 11Hypotheses about Correlations EPSY 640 Texas A&M University

  2. Hypotheses about Correlations • One sample tests for Pearson r • Two sample tests for Pearson r • Multisample test for Pearson r • Assumptions: normality of x, y being correlated

  3. One Sample Test for Pearson r • Null hypothesis:  = 0, Alternate   0 • test statistic: t = r/ [(1- r2 ) / (n-2)]1/2 with degrees of freedom = n-2

  4. One Sample Test for Pearson r • ex. Descriptive Statistics for Kindergarteners on a Reading Test (from SPSS) • Mean Std. Deviation N • Naming letters .5750 .3288 76 • Overall reading .6427 .2414 76 • Correlations • Naming Overall • Naming letters 1.000 .784** • Sig. (1-tailed) . .000 • N 76 76 • Overall reading .784** 1.000 • Sig. (1-tailed) .000 . • N 76 76 • ** Correlation is significant at the 0.01 level (1-tailed).

  5. One Sample Test for Pearson r Null hypothesis:  = c, Alternate   c • test statistic: z = (Zr - Zc )/ [1/(n-3)]1/2 where z=normal statistic, Zr = Fisher Z transform

  6. Fisher’s Z transform • Zr = tanh-1 r = 1/2 ln[1+  r  /(1 -  r |) • This creates a new variable with mean Z and SD 1/1/(n-3) which is normally distributed

  7. Non-null r example • Null: (girls) = .784 • Alternate: (girls)  .784 Data: r = .845, n= 35 • Z (girls=.784) = 1.055, Zr(girls=.845)=1.238 z = (1.238 - 1.055)/[1/(35-3)]1/2 = .183/(1/5.65685) = 1.035, nonsig.

  8. Two Sample Test for Difference in Pearson r’s • Null hypothesis: 1 = 2 • Alternate hypothesis 1  2 • test statistic: z =( Zr1 - Zr2 ) / [1/(n1-3) + 1/(n2-3)]1/2 where z= normal statistic

  9. Example • Null hypothesis: girls = boys • Alternate hypothesis girls  2boys • test statistic: rgirls = .845, rboys = .717 ngirls = 35, nboys = 41 z = Z(.845) - Z(.717) / [1/(35-3) + 1/(41-3)]1/2 = ( 1.238 - .901) / [1/32 + 1/38] 1/2 = .337 / .240 = 1.405, nonsig.

  10. Multisample test for Pearson r • Three or more samples: • Null hypothesis: 1 = 2 = 3 etc • Alternate hypothesis: some i  j • Test statistic: 2 = wiZ2i - w.Z2w which is chi-square distributed with #groups-1 degrees of freedom and wi = ni-3, w.= wi , and Zw = wiZi /w.

  11. Example Multisample test for Pearson r Nonsig.

  12. Multiple Group Models of Correlation • SEM approach models several groups with either the SAME or Different correlations: boys xy = a X y girls xy = a X y

  13. Multigroup SEM • SEM Analysis produces chi-square test of goodness of fit (lack of fit) for the hypothesis about ALL groups at once • Other indices: Comparative Fit Index (CFI), Normed Fit Index (NFI), Root Mean Square Error of Approximation (RMSEA) • CFI, NFI > .95 means good fit • RMSEA < .06 means good fit

  14. Multigroup SEM • SEM assumes large sample size, multinormality of all variables • Robust as long as skewness and kurtosis are less than 3, sample size is probably > 100 per group (200 is better), or few parameters are being estimated (sample size as low as 70 per group may be OK with good distribution characteristics)

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