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 Null hypothesis:  = 0, Alternate   0 test statistic: t = r/ [(1- r 2 ) / (n-2)] 1/2 with degrees of freedom = n-2

One Sample Test for Pearson r ex. Descriptive Statistics for Kindergarteners on a Reading Test (from SPSS) MeanStd. DeviationN Naming letters Overall reading Correlations NamingOverall Naming letters ** Sig. (1-tailed)..000 N7676 Overall reading.784**1.000 Sig. (1-tailed).000. N7676 ** Correlation is significant at the 0.01 level (1-tailed).

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

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

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/(35-3)] 1/2 =.183/(1/ ) = 1.035, nonsig.

Two Sample Test for Difference in Pearson r’s Null hypothesis:  1 =  2 Alternate hypothesis  1   2 test statistic: z =( Zr 1 - Zr 2 ) / [1/(n 1 -3) + 1/(n 2 -3)] 1/2 where z= normal statistic

Example Null hypothesis:  girls =  boys Alternate hypothesis  girls   2boys test statistic: r girls =.845, r boys =.717 n girls = 35, n boys = 41 z = Z(.845) - Z(.717) / [1/(35-3) + 1/(41-3)] 1/2 = ( ) / [1/32 + 1/38] 1/2 =.337 /.240 = 1.405, nonsig.

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

Example Multisample test for Pearson r Nonsig.

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

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

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)