Statistical tests and comparing variables

  • Decision on which statistical test to use for hypothesis testing depends on:
    • Type of data (continuous or categorical)
    • Whether the groups are independent or paired
    • Whether the data is normally distributed or not
    • Number of groups

CONTINUOUS OUTCOMES

  • Paired groups:
    • Repeated data on the same groups of patients e.g. before & after intervention

CATEGORICAL OUTCOMES

  • E.g. percentages or proportions
  • Will usually be shown in a table

CORRELATION AND REGRESSION

  • Correlation – concerned with strength (how close the points are to the straight line) & direction of association between variables 
  • Correlation coefficient – a quantitative measure, ranging from -1 to +1, of which the extent to which points in a scatter diagram conform to a straight line
  • Regression – demonstrates gradient (to what degree output [y] will change when input [x] changes) & direction of an association between variables
  • Regression coefficient – the parameters (i.e. the slope and intercept in simple regression) that describe a regression equation
  • Use scatter plots to aid examination between relationship between variables

Correlation

  • Correlation analysis is concerned with strength (how close the points are to the straight line) & direction of association between variables
  • Does not matter which variable is on the x & y axis as it does NOT infer causation
  • Calculates correlation coefficient, ranging from 1  0  +1 which indicates strength and direction of association
    • Negative – Y goes down as X goes up
    • No association (0) – no relationship
    • Positive – Y goes up as X goes up
  • Types of correlation tests:
    • Pearson’s correlation
      • Use only If the data is parametric (normally distributed)
      • Sensitive to extreme outliers
    • Spearman’s correlation
      • If the data is non-parametric (not normally distributed/skewed)
  • Disadvantages
    • Does not indicate magnitude of relationship
    • Can only compare 2 variables
    • You cannot calculate the correlation coefficient in these circumstances:
      • When the relationship is not linear
      • In the presence of outliers

Regression

Types:

  • Continuous data with linear relationship —> linear regression
  • Multiple variables –> multivariate regression
  • Binary outcome —> logistic regression
  • Survival —> cox regression

  • Linear regression
    • To define the relationship between variables & allows prediction of information
    • Establishes magnitude/gradient& direction of relationship = if X (input) ↑ by 1, predicts how that will affect Y (output)
    • Linear regression equation
      • y = mx + c
      • y is the output (outcome)
      • x is in the input (dependent variable)
      • c is the y axis intercept
      • m = slope of the straight line

In the graph above – the red dots are observations. The green lines represent random variability/deviations and the blue line represents the actual true relationship between the outcome (y) and the dependent variable (x). For example if x was number of fruits/veg. eaten per day in the third trimester and y was the birth weight in pounds of the baby. This graph would show a linear relationship between the two. The more fruits/veg per day, the heavier the baby. (completely made up example!!)

  • Fit statistics (R2)
    • Tells us strength of relationship from regression model
    • For a 2 variable linear regression – R2 is the same as the pearson’s correlation squared
    • Ranges from 0 –> 1 (does not give direction)

  • Multivariate regression
    • Allows multiple predictor/X variables
    • Adjusts for/controls for or removes effects of confounding factors
    • y = m1x1 + m2x2 + m3x3 + m4x4 ….. + c

  • Logistic regression
    • Outcome is binary (2 categories – yes/no)  
    • Probability of outcome = proportion of yes (changes binary data to number – 0.2 or 20%)
    • Proportion = P = ranges from 0–>1
    • Odds = p/(1-p) = ranges from 0 to infinity
    • Log odds = ranges from -infinitiy to +infinity
    • Allows us to do modelling/linear regression on a binary outcome
    • The anti-log/exponent of the log odds is the odds ratio
      • Explains how the probability of Y changes for a 1 unit increased in X
      • As this is a RATIO – no effect =1 OR, <1=decreased probability with increasing X, OR>1=increased probability with increasing X
      • E.g. cancer occurrence = exposure to asbestos (weeks)
      • OR=1.2 –> 20% more likely for cancer to occur for every week exposed to asbestos

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