Statistics Flashcards
Formula for standard error of the mean (SEM)?
SEM = SD / square root on (n)
SD - standard deviation
n = sample size
SEM gets smaller as sample size (n) increases
Definition of power of a study?
Power = 1 - the probability of type II error
The probability that a statistically significant difference will be detected
Probability of (correctly) rejecting the null hypothesis when it is false OR
Probability of confirming the alternative hypothesis when the alternative hypothesis is true
Power can be increased by increasing the sample size
Examples of observational studies
Cohort study
Case-control study
Cross-sectional study
Case series
Studies organised in level of evidence they provide.
Systematic reviews RCTs Cohort studies Case-control studies Cross-sectional studies Case series
Prospective cohort study
Sample recruited from population in the present, relevant predictors are measured, cohort is followed overtime to measure outcomes
Usual outcome measure is relative risk
Pro: more control over what is measured and how; can measure confounders
Con: expensive; wait until outcome occurs; rare outcome = need more participants
Retrospective cohort study
Cohort assembled after an outcome has occurred using stored data
Pro: cheaper, faster
Con: data quality limited
Case-control study
Start off with people with the disease and ask about exposure
Usual outcome measure is odds ratio
Pro: efficient for rare diseases and outbreaks
Con: hard to find matched controls; recall bias; confounding
Cross-sectional study
Random sample of a population in a point in time.
Descriptive: prevalence of a disease or exposure
Analytic: examine relationship between between different things e.g. obesity and arthritis
Can provide evidence of association but not about causality (hard to determine what came first)
Best study design for an intervention question?
Best primary study: RCT
Highest level of evidence: systematic review of RCTs
Best study design for question of harm or prognosis?
Prospective cohort study
Individual prospective cohort study
Retrospective cohort study
Case-control study
Best study type for questions of diagnostic test accuracy
Cross-sectional analytic study where the 2 tests are performed on the study participants
Best primary study type for prevalence of disease?
Cross-sectional descriptive study
Burden of disease
Best primary study type for incidence of disease?
Cohort study
Specified period of time; looks at cause of disease
Relative risk
The risk of something occurring relative to the chance of it occurring under different circumstances
= (incidence in exposed)/(incidence in unexposed)
i.e. use division
RR <1: treatment is beneficial
RR >1: treatment is harmful
RR = 1: treatment has no effect
Used in RCTs and cohort studies - need to know incidence
Absolute Risk Reduction
= (incidence of disease in exposed) - (incidence of disease in unexposed)
i.e. Use subtraction
Must remain aware if exposure has increased or decreased the risk
Number needed to treat
Number of people that need to be treated in order to prevent one negative outcome
NNT = 1 / (risk difference)
Odds Ratio
= (odds of exposure to the risk factor of interest in the cases) / (odds of exposure to the risk factor of interest in controls)
Used in case control studies
OR 0.6 = the exposed group is 40% less likely to develop specific outcomes compared to the control group
OR 1.5 = risk increased by 50%
P value
Probability that the observed results of the study are due to chance rather than an actual effect
IF p<0.05, the probability of getting the results by chance alone is 5% (i.e. statistically significant)
Confidence intervals
Provides us with a range within which we would expect the true effect to lie
Wide CI = poor precision
Narrow CI = good precision
IF using RR
Random error
Chance
Gives results either side of the true answer with the mean of all results being close to the true answer
Narrow confidence interval = less random error
Systematic error
Bias
Differ in one direction from the truth
Internal validity
How likely it is that the results are are correct for the sample of participants being studied.
Selection bias impacts the internal validity of a study
External validity
How likely it is that the results will hold true for other settings
= generalisability of the study
State 2 principles of a confounder
- has to be associated both with the risk factor of choice and the outcome
- fits into the causal pathway between the risk factor and the outcome (i.e. intervening variable)
Biases the results
Effect modification
Where the risk factor or intervention acts differently in one group compared to another
E.g. UV exposure, increased risk of melanoma and skin type
Loss to follow up
Losses before randomisation: affect the generalisability of our study
Losses after randomisation: relate to risk of bias
Intention to treat analysis
Means that we analyse people in the groups that they were originally randomised to, regardless of what actually happens during the study
Pro: preserves the effect of randomisation
Con: dilutes power
Composite endpoints
Rather than looking at several outcomes separately a study will combine several outcomes into the one composite measure that is used as the outcome
Why are they used?
- smaller sample size required to show effect
- allows assessment of ‘net’ effect of intervention
Why does it matter?
- Outcomes of high clinical importance can be grouped with those of minor importance
- Overestimate benefit of intervention
What is a funnel plot?
Special graph produced to assess likelihood of publication bias; must have >10 studies
Point estimate of the effect (e.g. RR or OR) plotted against a measure of the study’s size or precision
True value down centre
- smaller studies = larger scatter
- larger studies = closer to the true value
Sensitivity
Portion of those WITH the disease who have a positive test (i.e. true positive)
Sensitivity = TP / (TP + FN)
SnNout
When a highly sensitive test (Sn)
Is Negative (N)
the disease is ruled out (out)
If you want to avoid false negatives choose a test with high sensitivity (negative result in a sensitive test = confident patient doesn’t have disease)
Specificity
The proportion of those without the disease who have a negative test (i.e. true negative)
Specificity = TN / (TN + FP)
SpPin
When a highly specificities test (Sp)
Is Positive (P)
The disease is ruled in (in)
If you want to avoid false positives choose a test with high specificity
Positive predictive value
Probability of disease in those who test positive
= (TP) / (TP + FP)
Higher prevalence = higher PPV, lower NPV
Lower prevalence = lower PPV, higher NPV
Negative predictive value
Probability of no disease in those who test negative
= TN / (TN + FN)
Higher prevalence = higher PPV, lower NPV
Lower prevalence = lower PPV, higher NPV
NPV / PPV depend upon the prevalence of the characteristic in a given population
Positive likelihood ratio
= (probability of a +ve test in those with the disease) / ( probability of a +ve test in those without disease)
i.e. sensitivity / 1-specificity
Larger PLR = greater likelihood of disease
PLR > 10 will be useful in ruling in disease
PLR = 1 indicates a useless test
Negative likelihood ratio
= (probability of -ve test in those with disease) / (probability of -be test in those without disease
i.e. (1-sensitivity) / specificity
Smaller NLR = lower likelihood of disease
NLR <0.1 will be useful in ruling out disease
NLR = 1 indicates a useless test
Bias in screening
Lead time bias - apparent longer survival in screen detected cases as identified at earlier point in disease
Length time bias - slowly progressive disease more likely to be picked up by screening
Level of evidence
Ia- evidence from meta-analysis of RCTs
Ib - evidence from at least one RCT
IIa - evidence from at least one well designed controlled trial that is not randomised
IIb - evidence from at least one well designed experimental trial
III - evidence from case, correlation and comparative studies
IV - evidence from a panel of experts
Grade A - based on evidence from at least 1 RCT
Grade B - based on evidence from non-RCT
Grade C - based on evidence from a panel of experts
Post test probability
Pre test probability = prevalence
Post test probability = prevalence x LR
Post test probability after a +ve test = prevalence x PLR
Post test probability = (post-test odds)/(post test odds + 1)
Post test odds = (pre-test odds) x (likelihood ratio)
Best estimate of prevalence?
Prevalence = incidence x duration
E.g. disease has annual incidence of 15 cases per 100,000. Mean survival after diagnosis is 5yrs.
Prevalence = (15 per 100,000) x 5 = 75 per 100,000
Type 1 error
Rejecting the null hypothesis when it is in fact true OR
Accepting the alternative hypothesis when it is in fact false (i.e. a false positive result)
p value = probability of a type 1 error
p value
Probability of a type 1 error OR
The probability of finding a difference when there is one
Significance is conventionally set at p < 0.5
Type 2 Error
= power
Accepting the null hypothesis when it is false
Observing no difference when there is one
A false negative result
Rejecting an alternative hypothesis when it is true
Power
= 1 - probability of Type 2 error
Likelihood of finding an effect when it is present
i.e. likelihood of avoiding false negatives
Main modifiers of power
- Size of effect
- more difficult to detect small effects - Sample size
- larger sample size = easier to detect effect - Desired significance
- i.e. p<0.001 will conclude fewer positive than p<0.05 - Standard deviation
Multivariate analysis
Used to determine whether or not confounding is occurring due to other factors
ROC curve
Y axis: true positive (sensitivity)
X axis: false positive (1-specificity)
Test with good performance swoops into the top L corner
Test close to a diagonal line is no better than chance at discriminating between those with and those without the disease
Student’s T-test
Parametric (normally distributed)
Paired or unpaired
Pearson’s product moment coefficient
Parametric
Correlation of 2 variables
Mann-Whitney U Test
Non-parametric
Unpaired data
Wilcoxon signed rank test
Non-parametric
Compares 2 sets of observations on a single sample
Chi squared test
Non-parametric
Used to compare proportions or percentages
Spearman, Kendall Rank
Non parametric
Correlation
Paired vs unpaired data
Paired data: obtained from a single group of patients e.g. measurement before and after an intervention
Unpaired data: 2 different groups of patients e.e. Comparing response to different interventions in 2 groups
Hazard Ratio
Similar to relative risk but used when risk is not constant in time.
Typically used when analysing survival over time
Reduction in risk of death or progression
HR of 0.84 = 16% reduction in risk
