Statistics Lecture 3: Probability and Sampling

Lecture 3: Probability and Sampling Probability: how likely an event is to occur -​ Quantified as a number between 0 and 1 -​ 0 = zero chance -​ 0.5 = even chance -​ 1 = certain to occur -​ The higher the probability the more certain the event is to occur -​ Probability = number of possibilities / total number of outcomes Sample Size and Population -​ As the number of people in the sample increases: -​ The mean of the sample gets closer to the mean of the population -​ The distribution becomes more normal* -​ Occurs around 30 observations/people Z-Scores -​ Also known as the ‘standard score’ -​ Z-score is a measure of how many SD above or below the mean -​ Ranges from -4 SD to +4 SD -​ Also tells you where a particular score is in relation to the mean -​ Matches group mean: z-score = 0 -​ Above group mean: z-score is + -​ Below group mean: z-score is -​ Z-scores allow comparison of observed distribution to expected distribution ie. if it is normally distributed -​ Scores that are abnormal are defined by those that occur less than 95% of the time (< -2SD and >+2SD) -​ Z = (score - mean score) / standard deviation -​ Outliers: in a normal distribution any measurement with a Z-score greater than +3 or less than -3 is called an outlier What is a Normal Distribution? -​ A visual representation of a group of scores (probability distribution) -​ Y-axis = number of people/frequency -​ X-axis = variable of interest (eg. weight, blood glucose etc.) -​ Characteristics of a normal distribution: -​ Bell-shaped curve -​ Most of the data close to the mean -​ Symmetrical -​ The tails come closer and closer to the x-axis but never touch -​ Total area under the curve is 100% -​ -​ -​ -​ Highest probability is in the center The numbers on the x-axis refer to the standard deviations from the mean The area under any portion of the curve is the probability of x being within that span The 68-95-99.7% Rule: -​ +/- 1 SD = about 68.2% -​ +/- 2 SD = about 95.45% -​ +/- 3 SD = about 99.7% Z-Score and Central Limit Theorem -​ If a sample is large enough the distribution of probable outcomes takes the shape of the normal distribution, sometimes called the bell curve = CENTRAL LIMIT THEOREM -​ The more people in the sample, the more the mean of the sample will “look like” the wider population -​ The variance decreases -​ The less variance means more normal distribution Sample Size -​ The larger the sample, the more precise -​ Sample too small = unable to detect a difference -​ Sample too large = may find differences that don’t exist -​ Calculating sample size includes: (don’t worry we don’t do calculations) -​ Significance level (a) - usually a = 0.5 -​ Smallest effect size interested in -​ Standard deviation -​ Power - the probability of finding a significant difference (usually 80%) -​ The sample needs to be homogenous, because if it is too different you cannot make solid inferences about the data -​ If too homogenous it may limit generalizability: -​ Inclusion criteria -​ Exclusion criteria -​ Sample size is determined by the effect that you want to see = Cohen’s d (standardized measure of difference between 2 groups) -​ Small = <0.3 -​ Medium = 0.3-0.5 -​ Large = >0.5 -​ Sample size and effect size are inversely related (the smaller the effect size the larger the sample that is required to see it) Types of Sampling 1.​ Probability Sampling a.​ Simple random sampling i.​ ii.​ iii.​ Randomly select participants Everyone as an equal chance of being selected Population must be accessible (often not feasible due to money/resources/time) b.​ Systematic i.​ Pick a random starting point and select every ‘n’th person c.​ Stratified i.​ Divide population into characteristics of population important for the research ii.​ Sample is matched to the proportion within the population d.​ Cluster i.​ Sample is the whole group ii.​ Entire population may not be known iii.​ Basis of cluster can be location, GP practice 2.​ Non-probability Sampling a.​ Convenience i.​ Collect data from the easiest available group ii.​ Non-random iii.​ May not be representative b.​ Quota i.​ Like stratified random sampling but participants are not randomly selected ii.​ Continually recruit within each group until required sample size is obtained Sampling Error or Bias? -​ No matter which sampling method you use, there will always be sampling error due to chance alone -​ Random sampling avoids the most bias* -​ Bias is a systematic error made when using a random sampling approach that results in a nonrandom sample -​ Error is random and can be based on the research design, and is often out of your control -​ Many types of bias: selection bias, participation bias, measurement bias, performance bias, reporting bias Sampling Bias: sampling bias occurs when some members of the population are systematically more likely to be selected in a sample than others Error: always present in a sample (unavoidable) Effect Size: it is a statistical concept that measures the strength of the relationship between 2 variables on a numeric scale Confounding Variable: a third variable in a study that is related to both the supposed cause and effect of the study. Confidence Intervals -​ A confidence interval communicated how accurate our estimate (from the sample) of a population parameter is likely to be -​ The width of the confidence interval depends on 2 things: -​ The variation within the population of interest (more variation in population = more variation and wider intervals) -​ The sample size (large sample = more narrow interval) -​ Confidence intervals are used to calculate the boundaries within which we believe 95% of the population will fall