A coworker called me a numbers person yesterday. This was shocking because I am not (I don't think)... but, if you are going to take on the BCPS certification, statistics will be about 25% of the test (or so they say). Guess what? After studying and learning stats as it pertains to my field (medical studies and pharmacy) I can say that I may be a lover of stats now. This statement would not have been uttered much less thought in April 2012 when I started this journey.
Background and the Endpoint:
- We can’t study every single person. We draw conclusions or make inferences to the data so that we can determine if we are going to accept or reject the null hypothesis. There is no difference between the two groups (null hypothesis).
- Known as the arithmetic mean - parametric, continuous data
- Caution with outliers; may need to present data as median
- Middle most value (or) 50th percentile value
- Usually used with ordinal data or continuous data not normally distributed (outliers that skewed the data)
- Most frequently occurring number
Mean = Median = Mode (roughly equal)
Range = Interval between the lowest and highest value
Interquartile Range = 25th - 75th percentiles and related to the median
Standard Deviation (+/- SD):
- SD = square root of variance
- Estimates degree of scatter of sample data points about hte sample mean; only applicable to parametric data
Standard Error of the Mean (SEM):
- SEM = SD/square root of n (sample size)
- Average variability within a sampling distribution (if study repeated?)
- Caution in its use and interpretation; smaller than SD
Confidence Interval: 95% confident that the data falls in this area (want tighter) -- make this tighter by increasing the sample size.
Types of Data Scales:
- Continuous - (Parametric) - follow assumption that the median = mode = mean (follow normal distribution (Common statistical tests: T-test, ANOVA)
- Nonparametric - Nominal data, Ordinal data, continuous data (not meeting certain assumptions) (Common stat tests: Mann-Whitney U, Wilcoxan Rank Sum, Sign Test)
Positively Skewed Data - Skewed to Right (refers to most data points to the left and the tail to the right) (Mean > Median > Mode)
Negatively Skewed Data - Skewed to the Left (refers to most data points to the right and the tail to the left) (Mean < Median < Mode)
- Categorial variables that have no sense of “ranking”
- Only get a mode and frequency
- No measure of variability to describe this type of data (i.e., standard error of the mean, SEM; or standard, SD)
- Numbers are arbitrarily assigned to characteristics for data collection
- Examples: Gender (yes or no), Mortality (yes or no), Continuous data can also presented as a nominal endpoint so be careful how it is asked. “How many people achieved SBP of < 140” -- trick question of using a continuous data type as NOMINAL by the way the question was asked (meeting a SBP goal or cut-off).
- Can generate a mode, median, and frequency
- Still cannot use SEM and SD
- Measure of variability: Range, interquartile range
- Numbers used to indicate rank-order, but do not have the same magnitude of difference between them
- Examples: NYHA FC for heart failure, Pain scale, Glasgow Coma Scale
- Examples of trickery: Blood pressures into groups - 100-120, 120-150, 150-190 treating the continuous points as ordinal data
- Can get a mode, median and mean
- Measure of variability - standard deviation and range, interquartile range
- Interval Data - Units of equal magnitude, rank order but is without an absolute zero (temperature)
- Ratio Data - Same as interval, but there is absolute zero (e.g., pulse or blood pressure)
- Not the same pts in each group (i.e., they are different people, though they may be matched based on certain characteristics)
- Groups being studied are not different
- Examples: patient in a cross-over study, identical twins, right eye vs left in the same pt
Fisher’s Exact is used in SMALL sample sizes (<30-40) over Chi-square (Nominal, two independent sample tests)
Mann Whitney U for continuous when the continuous data doesn’t meet the assumptions of being parametric or follow normal distribution. (i.e., outlier)
Hypothesis Testing - Power Analysis
- Power = 1 - Beta
- Indicates the probability that a statistical test can detect a significant difference when it in fact it truly exists
- Since Beta indicates the probability of making a type II error, the power calculation tells you the probability that you will NOT make a type II error.
- If you reject the null hypothesis, low power not enough patients?
alpha = 0.5 - 5% chance that the results that you find are not right. The smaller your alpha the least likely to make a Type I Error.
Power Analysis - Accept or Reject Null Hypothesis Decision -- Reality Null False or True
Factors that determine or impact the Power of a study:
- Alpha value (alpha)
- Sample size (n)
- Types of groups evaluated in the study
- Type of alternative hypothesis selected
- Choice of statistical test used
Components needed for power calculation:
- Sample size (n)
- Standard deviation
- Practical significance (Delta)
- Z alpha (1-sided is 1.65 and 2-sided is 1.96)
- Z beta (based on level of power chosen)
Alpha (alpha) Value
- Should be determined a-priori (prior to test)
- Defined as the max acceptable probability of making a type I error
- Most will not select alpha > 0.05 or 5% or < +/- 2 SD from the mean
- The alpha and sample size have greatest impact on the power of a study
- WILL NOT tell you the degree of clinical significance
- It helps to determine the likelihood that a difference seen is due to chance or random error
- A p<0.0001 is NOT more “clinically significant” than a p<0.001
- P=0.01 means that there is a 1% chance that the results you found in your statistical analysis are either due to chance or random error.
- The smaller the p-value the more likely the results you found are real; that is why a researcher wants small p-values!!!
95% Confidence Interval
- What does it really tell you?
- This depends on the SEM and the degree of confidence we choose (95% vs. 99% CI)
- 2 +/- SD is 95% of data; 3 +/- D is 99%
- To calculate a CI you need the: mean, SD, sample size (n) and Z-score (which is 1.96 for 95% CI and 2.58 for 99% CI)
- The closer a point lies to the middle of the 95% CI the more likely it will reperesent the true population
- The CI can be made to be more narrow or precise with an increase in sample size (n)
- Interpreting the significance in a given CI
- 95% CI be rejected for Odds Ratio or Relative RIsk if it contains “1.0”
We are trying to avoid making a:
Type I Error (the results say there is a difference but in reality there is no difference)
Type II Error (the results say there is no difference but in reality there is a difference)
Relative and Absolute Risk and Odds Assessment
Relative Risk (RR): the risk of the event after the experimental treatment as a percentage of the original risk
RR = incidence rate in exposed patients / incidence rate in non-exposed patients
RR = 1 (incidence is the same for both groups)
RR = >1 (incidence in exposed group is higher)
RR = <1 (incidence in exposed group is less)
RR = 0.33/0.24
Nominal data - Chi Square
ERR = 37.5% excess risk in causing edema
RR > 1 excess risk of something to the exposed group - in this case the CCB causes edema
“37.5% excess risk in causing edema”
RRR - Relative Risk Reduction when the RR is < 1
RRR = 1-0.581 = 0.419 = 41% of risk for an MI was decreased from use of ASA
Nominal data - Chi square
Absolute Risk Reduction
ARR = RR control - RR treatment
=(248/3,293) - (147/3,275)
=0.0226 or 2.26
NNT = 1/ARR
=1/0.0226 = 44 pts
So I would have to treat 44 pts for 4.9 years to prevent 1 MI or death from CHD
Sample RR Interpretation:
The effect of ASA on vascular death relative to no aspirin was a RR of 0.8 and the RRR was 20%. How should we interpret that?
The risk of vascular death in the aspirin group at 5 weeks is 80% of the risk in the control group, therefore ASA reduced the risk of vascular death by 20%
Estimates the RR in retrospective studies, since the # of pts at risk is not known, preventing the calculation of incidence.
Use of OR in prospective study will overestimate the risk!
When incidence of isease is
OR = (AD)/(BC)
OR = 8.88 means that 8 times more likely to have an MI with cocaine use!
Correlation Coefficient (r)
- r = -1 to +1
- The strength of the “relationship” between 2 variables whereas regression analysis provides the “predictability” of one variable on another variable.
- Correlation does NOT imply causation
- The closer the absolute value is to +1, the stronger the “correlation” or “relationship” between the 2 variables.
- The closer to -1 = no relationship
How to interpret an “r” value:
- Though a “strong association (i.e., r= close to +1) may exist between 2 variables, it does not “imply” that one event or variable “caused” another.
- The “r” value does not have the ability to determine which came first
- Another variable may also be influencing the relationship
- Most complex relationships can rarely be explained by only 2 variables
- Thus, Spearman and Pearsons Correlations are only descriptors of the “strength” and “direction” of the relationship
Coefficient of variation (r-squared)
- How much one variable can be explained or related to another
- also called coefficient of determination
- Example related to number of calories and weight gain: If r=0.84, then r squared = 0.70; therefore 70% of the variation of Y is due to X
- This implies that 70% of the variability in wt gain or loss is attributed to the variability in the amount of cals consumed (or) it can also be said that the amount of calories consumed provides us with 70% of the information needed to predict weight gain or loss.
- Mathematical description or equation that provides the “predicctability” of one variable on another variable
- Provides “statistical control” of confounders
- Linear Regression - Relationship between 2 variables (1 independent and 1 dependent variable) - Uses line of best fit (y = mx + b)
- Multilinear Regression - Relationship between > 1 independent variable and 1 dependent variable
- Logistic Regression Analysis - which variables are important in predicting the outcome
- A variable may not always be a significant influence on the outcome:
- Main reason is no reln exists between the ind and dep variable
- One variable is highly correlated with another variable already included in the regression and does not provide additional protective information
- Lack of statistical power to detect a difference (usually from insufficient sample size).
- A variable may not always be a significant influence on the outcome: