Examining the Relationship Between Two Variables (Bivariate Analyses)

What type of analysis? We have two variables X and Y and we are interested in describing how a response (Y) is related to an explanatory variable (X). We have two variables X and Y and we are interested in describing how a response (Y) is related to an explanatory variable (X). What graphical displays do we use to show the relationship between X and Y ? What graphical displays do we use to show the relationship between X and Y ? What statistical analyses do we use to summarize, describe, and make inferences about the relationship? What statistical analyses do we use to summarize, describe, and make inferences about the relationship?

Type of Displays Y is Continuous Scatterplot Comparative Boxplot Y is Ordinal or Nominal Logistic Plot 2-D Mosaic Plot X is Continuous X is Ordinal or Nominal

Fit Y by X in JMP X Variable/Predictor Data Type Y Variable/Response Data Type In the lower left corner of the Fit Y by X dialog box you will see this graphic which is the same as the more stylized version on the previous slide.

Type of Displays Y is Continuous Scatterplot Comparative Boxplot Y is Ordinal or Nominal Logistic Plot 2-D Mosaic Plot X is Continuous X is Ordinal or Nominal

Type of Analyses Y is Continuous Y is Continuous Correlation and Regression - Parametric or Nonparametric If X has k = 2 levels then Two-Sample t-Test or Wilcoxon Rank Sum Test. If X has k = 2 levels then Two-Sample t-Test or Wilcoxon Rank Sum Test. If X has k > 2 levels then Oneway ANOVA or Kruskal Wallis Test If X has k > 2 levels then Oneway ANOVA or Kruskal Wallis Test Y is Ordinal or Nominal Y is Ordinal or Nominal If Y has 2 levels then use Logistic Regression If Y has 2 levels then use Logistic Regression If Y has more than 2 levels then use Polytomous Logistic Regression If Y has more than 2 levels then use Polytomous Logistic Regression If both X and Y have two levels then use Fisher’s Exact Test, RR/OR, and Risk Difference/AR If both X and Y have two levels then use Fisher’s Exact Test, RR/OR, and Risk Difference/AR If either X or Y has more than two levels use a Chi-square Test. If either X or Y has more than two levels use a Chi-square Test. McNemar’s Test (dependent) McNemar’s Test (dependent) X is Continuous X is Ordinal or Nominal

Fit Y by X in JMP X continuous X nominal/ordinal Y nominal/ordinal Y continuous

Example: Low Birthweight Study (Note: This is not NC one) List of Variables id – ID # for infant & mother id – ID # for infant & mother headcir – head circumference (in.) headcir – head circumference (in.) leng – length of infant (in.) leng – length of infant (in.) weight – birthweight (lbs.) weight – birthweight (lbs.) gest – gestational age (weeks) gest – gestational age (weeks) mage – mother’s age mage – mother’s age mnocig – mother’s cigarettes/day mnocig – mother’s cigarettes/day mheight – mother’s height (in.) mheight – mother’s height (in.) mppwt – mother’s pre-pregnancy mppwt – mother’s pre-pregnancy weight (lbs.) weight (lbs.) fage – father’s age fedyrs – father’s education (yrs.) fnocig – father’s cigarettes/day fheight – father’s height lowbwt – low birth weight indicator (1 = yes, 0 = no) mage35 – mother’s age over 35 ? (1 = yes, 0 = no) smoker – mother smoked during preg. (1 = yes, 0 = no) Smoker – mother’s smoking status (Smoker or Non-smoker) Low Birth Weight – birth weight (Low, Normal) Continuous Nominal

Example: Low Birthweight Study (Birthweight vs. Gestational Age) Y = birthweight (lbs.) Continuous X = gestational age (weeks) Continuous

Regression and Correlation Analysis from Fit Y by X

Example: Low Birthweight Study (Birthweight vs. Mother’s Smoking Status) Y = birthweight (lbs.) Continuous X = mother’s smoking status (Smoker vs. Non-smoker) Nominal

Independent Samples t-Test from Fit Y by X

Example: Low Birthweight Study (Birthweight Status vs. Mother’s Cigs/Day) Y = birthweight status (Low, Normal) Nominal X = mother’s cigs./day Continuous P(Low|Cigs/Day)

Logistic Regression from Fit Y by X

Example: Low Birthweight Study (Birthweight Status vs. Mother’s Smoking Status) Y = birthweight status (Low, Normal) Nominal X = mother’s smoking status (Smoker, Non-smoker) Nominal

Independent Samples p 1 vs. p 2 - Fisher’s Exact, Chi-square, Risk Difference, RR, & OR Skipped the arrows this time, everything should self-explanatory. Notice the OR is upside-down and needs reciprocation. OR = 1/.342 = 2.92

Summary In summary have seen how bivariate relationships work in JMP and in statistics in general. We know that the type of analysis that is appropriate depends entirely on the data type of the response (Y) and the explanatory variable or predictor (X).