Analyzing Bivariate Data: Interval/Ratio. Today s Content

Transcription

1 Analyzing Bivariate Data: Interval/Ratio Day April 26 C. Zegras Today s Content Understanding your data: Exploration Means of exploring bivariate data Looking at bivariate relationships: Correlation Understanding and calculating the correlation coefficient Guidelines for using the correlation coefficient 1

2 Basic Guidelines: Exploring your Data Before you begin quantifying anything 1. Think. What would you expect in terms, e.g., of dependent and independent (or response and explanatory) variables 2. Plot your data; add numerical summaries 3. Look for overall patterns and any deviations 4. When regular patterns emerge, look for parsimonious means of presentation 2

3 The Scatterplot 1 US.9.8 Norway/Iceland Luxembourg Human Development Index Equatorial Guinea Sierra Leone.1 UN, HDR, GDP per Capita (PPP US$21) Interpreting the Scatterplot Look for overall patterns and major deviations from that pattern Particularly, outliers Understand the form, direction, and strength of relationship 3

4 Association Positive association between two variables Above-average values of one tend to accompany above-average value of the other and below-average values also tend to occur together. Negative association. Above-average values accompany belowaverage values, and vice versa. The strength of the association can be inferred from adherence to a clear form Human Development Index HDI-High HDI-Medium HDI-Low GDP pe r Capita ( PPP US$21) UN, HDR, 23 4

5 Correlation Describes the strength and direction of a linear relationship between two variables 5

6 Understanding the calculation of r Formally (or originally), Pearson s productmoment correlation coefficient r, the correlation coefficient represents the standardized covariance (or co-movement ) between two variables, X and Y Formally: r = n n n n X Y i i X i Y i i = 1 i = 1 i = n n n n n 2 X i X i n Y 2 i Y i i = 1 i = 1 i = 1 i = 1 More practically, computing r 1. Find the mean and standard deviation for the variables x and y: x, y, s s x, y 2. Use the means and standard deviations to calculate the standardize scores for each x- value and each y-value 3. The correlation is the average of the products of the standard scores Steps 2 and 3: r = 1 x x y y n 1 s x s y 6

7 Example: Activity Levels and Health Observation (Adults) Activity Levels (minutes of moderate intensity physical activity per day) Body Mass Index (BMI) (weight in kg / [height in m] 2 ) Which variable should we label x and which y? Scatterplot of BMI and Activity Levels? BMI Activity Levels (minutes) 7

8 l Let s calculate r Observation Activity Levels Body Mass (Adults) (minutes of moderate Index (BMI) intensity physical (weight in kg / activity per day) [height in m] 2 ) Important points about r It describes strength of linear relationship between two variables Look at a scatterplot of your data first Human Deve opment Index HDI-High HDI-Med ium HDI-Low GDP per Capita ( PPP US$21) 8

9 Important points about r It is not dependent on the particular units of measurement of your variables E.g., whether you measure weight in pounds or kilograms Why is that? Computer programs will calculate r for any pair of variables Irrespective of appropriateness of the data Again: Check your data first! Important points about r Basic law of statistics: Correlation is not causation! Cannot tell the difference between x and y Does not prove a theory Does not account for your research design Common Response Confounding X X Z Z Y Y 9

10 In our simple example of BMI and activity levels: potential lurking variables At the same BMI, women tend to have more body fat than men. At the same BMI, older people, on average, tend to have more body fat than younger adults. Highly trained athletes may have a high BMI because of increased muscularity rather than increased body fatness. Important points about r r is vulnerable to outliers use it carefully in such cases Test without outliers r = r = Human Development Index Human Development Index HDI-H igh.8 HDI-High GDP per Capita ( PPP US$21) GDP per Capita ( PPP US$21) 1

11 In Summary R-values can only be computed for interval or better data. Remember: Correlation is not causation (1) we are only showing some level of a linear association between data, and (2) we may have a spurious relationship in which common response or confounding variables cloud the true story. Remember: causation cannot be proven using statistical analysis. Intuitively, you should know that the r-value is equal to the sum of the products of the z-scores for both variables, divided by n-1. Remember: the r-value only relates to linear relationships (many relationships in the social sciences are not linear). 11