biom301 mod 3 - describing & presenting bivariate data

bivariate data

2 variables measured on same EU independently and without bias

2 qualitative

• contingency table

• side by side bar graphs

• side by side circle graphs

2 quantitative

• scatter plots

1 qualitative & 1 quantitative

• side by side box & whisker plot

• side by side stem & leaf 

• side by side frequency histograms

linear correlation

a linear relationship between 2 quantitative variables

• only use correlation if LINEAR

correlation coefficient

( r ) - shows strength of relationship (positive/negative)

• “r” measures direction & strength of linear relationship between 2 variables

• insignificant if close to zero

• miss real relationships if it’s not linear

• ALWAYS shown by scatter plot

Correlation or not?

• no correlation if “y” does NOT change when “x” changes (r = 0)

  • or when “y” changes but “x” doesn’t

• positive correlation if “y” increases when “x” increases (r = +)

  • r = +1 → perfect positive correlation

• negative correlation if “y” decreases when “x” increases (r = - )

  • r = -1 → perfect negative correlation

• -1 < r > +1 → intermediate relationship

correlation concerns

• check for nonlinear relationships 

  • transform data to fit linear model by using a nonconstant

• check for outliers

  • less ‘tight’ r value (bigger)

  • need justification to remove valid data from dataset

• correlation is NOT causation

  • most correlations are done on survey data

    • surveys CANNOT determine cause & effect

• third-variable problem

  • 2 variables could have strong correlation, but b/c of a 3rd “lurking” variable

• never predict/extrapolate beyond your data set

terms to describe association

• associated

• tends to

• linked → trends

• connected

• tied to

regression

asks if changes in 1 variable cause or predicts changes in another variable

• can be a curve (not restricted to linear relationships)

linear regression

predict a value for y (output/dependent variable) given a value of “x” (input/independent variable)

• determines line of best fit

• for any value of “x”, you can predict a value of “y” given regression equation

line of best fit

the linear trend that best fits/describes the data set

• minimize deviations between line & actual data points vertically

  • b/c trying to predict “y” value

2 components for best fit line equation

1. estimate of linear slope ( b₁ )

2. estimate of intercept ( b₀ )

regression coefficient

( R² ) - the amount of variability in the dependent variable (y) explained by the variability in the independent variable (x)

• R² = 0 → no relationships between x & y

• R² = 1 → perfect relationship/straight line

R² vs r

r → correlation, tightness, direction

• –1 < 0 > +1 

• Can statistically test for relationship between x & y

• Looks for trends in 2 quant variables, linear relationship, usually survey data

R² → regression, tightness ONLY

• 0 < R² > 1

• Not tested statistically 

• Asking if y variables is a function of the x variable 

• Can deal w/ 2+ variables, curvilinear data, both survey & experiment

  • Causation requires controlled experiment 

• Tells how much of the variability in y is explained by the x variable 

regression test

test for significant slope

• if slope of line ( b₁ ) is statistically significant different from zero

• if intercept ( b₀ ) is significantly different from zero

slope question

• Slope is NOT significant diff from zero → y does not change as a function of x

• Slope IS significant diff from zero → y decreases as a function of x

components of regression graphs

• title

• labeled axes

• line ONLY in the range of data

• equation of line

• R² value

regression concerns

• Outliers can have big impact

• Never extrapolate beyond range of data

• Relationship may be nonlinear (graph first to be sure) 

• Lurking variables if survey data 

• Interpretation