Correlations 2 Real Example 1. Speed limits on reads - For every 1mph the average speed of vehicles reduces on roads, there are 6% fewer accidents (The Department of Transport) Accidents = 6 vehicle speed + a Driver groups claim this relationship is not causal - speed limit reduction is confounded with better (safer) cares and better (safer) roads. For every 20,000 cards on the road there is one road death Road deaths = 20,000 vehicles + a Correlation between number of miles driven by lorry drivers and accidents - Insurance is more for driving above 16,000 miles per year 2. The Flynn effect - IQ appears to be increasing IQ = 0.31 Year - 553.220 For every year, IQ goes up by 0.31 points 3. Correlation between building skyscrapers and financial crashes. Financial crash = B skyscrapers built + constant 4. Negative correlation between amount of investment in NHS and waiting time for operations. Waiting time = constant - B investment Hypothesis Testing Null hypothesis - any relationship found is due to chance Experimental hypothesis - 1. nonffdirectional (two tailed) there will be a relationship. 2. Direction (one tailed) there will be a positive relationship between two variables/there will be a negative relationship between two variables Significance tests: Parametric versus nonffparametric. Both give a p value (how likely the relationship was to occur due to chance depending on the size of N). Alpha - the level at which we decide the relationship is not just due to chance/usually .05. But this is doubled to .10 for onefftailed results. P <. 05 is statistically significant meaning we can reject the null hypothesis. Two Types of Correlation Spearman's Rho (Non-Parametric Correlation) Nonffparametric test used to measure the strength of association between two variables, where the value r = 1 means a perfect positive correlation and the value r = ff1 means a perfect negative correlation ffUsed with Ordinal Data - X and Y values are ranks. ffUsed with interval data that does not meet the parametric assumptions. E.g. we want to estimate whether a teacher's impression of their student's intelligence (on a scale from 1ff 10) are related to the students' actual intelligence.
Spearman's Rho Logic: Rank the sets of numbers (these ranks can be identical which would create a Spearman Rho of +1) These ranks can be opposite (which would create a Spearman Rho of ff1) Dealing with ties: Share the rank of the tied values. Data 1 3 4 6 1 Rank 2nd 3rd 4th 1 st Data 2 99 103 106 72 16 Rank 2nd 3rd 4th Data 1 4 4 2.5° 6 Rank 2.5th 4th The likelihood of getting a particular rank order depends on the size of the sample (N) Data 1 1 8 6 2 1st Rank 2nd 3rd 4th Data 2 110 85 83 112 4th Rank 3rd 2nd 1 1 st Calculating the Spearman Rank Coefficient: Exam Grade Rank Anxiety Rank Dif Dif Sq 40 3 86.3 65 7 88.72 80 9.5 70.18