This document are a summary of rules for self-reference from the book Statistical Rules of Thumb by Gerald van Belle. Most of the notes have been taken verbatim. Please refer to the book for a detailed description without which the notes may be meaningless to the uninitiated reader.

# 1 The Basics

1. Any statistical treatment must address the questions
• What is the question?
• Can it be measured?
• When, where, and how will you get the data?
• What do you think the data are telling you?
1. Observation is selection

2. Replicate to characterize random variation

3. Variability occurs at multiple levels

4. Invalid selection is the primary threat to valid inference

5. Compared with experimental studies, observational studies provide less robust information

6. Make a sharp distinction between observational and experimental studies

7. Always look for a physical model underlying the data being analyzed. Assume that a statistical model, such as a linear model, is a good first start only

8. Keep models as simple as possible but no more simple

9. Be sure to understand the components and purpose of an omnibus quantity

10. Do not multiply probabilities more than necessary. Probabilities are bounded by 1; multiplication of enough probabilities will always lead to a small number

11. The use of one sided p-values is discouraged. Ordinarily, use 2-sided p-values

12. When designing experiments or observational studies, focus on p-values to calculate sample size; when representing results, focus on sample size

13. Use atleast 12 observations in constructing a confidence interval

14. For samples $$\geq$$ 20, a point estimate +/- 2 standard errors has a 95% coverage for a wide variety of distributions

15. Always know what the unit of a variable is

16. Do not let scale of measurement rigidly determine method of analysis

17. The practical applied statistician uses methods by all three schools (Neyman-Pearson, Likelihood, Bayesian) as appropriate

# 2 Sample Size

1. The basic formula (Lehr’s equation) for sample size is $n = 16/\Delta^2$ where $\Delta = \frac{\mu_0 - \mu_1}{\sigma} = \frac{\delta}{\sigma}$ is the standardized difference. In the single sample case (where a single sample is compared to a known population value), the numerator is 8 instead of 16

2. The sample size using coefficient of variation (CV) is given by $n = \frac{16(CV)^2}{(ln(\mu_0)-ln(\mu_1))^2}$

3. Finite population size correction can be ignored in initial discussions of survey sample size questions

4. The range of the observation is related to the standard deviation as follows: $\frac{range}{\sqrt{2(n-1)}} \leq s \leq \frac{n}{n-1}\frac{range}{2}$

5. Do not formulate objectives for a study solely in terms of effect size

6. Confidence intervals associated with statistics for two variables can overlap as much as 29% and the statistics can still be significantly different

7. If $$\theta_1$$ and $$\theta_2$$ are the means of two poisson-distributed populations, then the required number of observations per sample is $n = \frac{4}{(\sqrt(\theta_1)-\sqrt(\theta_2))^2}$

8. The sample size calculation for a poisson distribution with background rate $$\theta*$$ is given by $n = \frac{4}{(\sqrt(\theta* + \theta_1)-\sqrt(\theta* + \theta_2))^2}$

9. The sample size calculation for a binomial distribution is given by $n = \frac{16\bar{\pi}(1-\bar{\pi})}{(\pi_0 - \pi_1)^2}$ where $\bar{\pi}=\frac{\pi_0 + \pi_1}{2}$

10. For unequal sample sizes where one group contains $$n_0$$ samples and the other group contains $$kn_0$$ samples, choose k such that $k = \frac{n_0}{2*n_0-n}$ to get the same precision as having an equal number of samples in each group

11. When there are different costs associated with each sample, choose a sample size that is inversely proportional to the square root of the cost of the observations

12. Given no observed events in $$n$$ trials, the 95% upper bound on the rate of occurence is $$3/n$$

13. Sample size calculations should be based on the statistics used in the analysis of the data

# 3 Observational Studies

1. The model for an observational study is the sample survey

2. Large sample size do not guarantee validity

3. Good observational studies are designed

4. To establish cause and effect requires longitudinal data

5. Make theories elaborate. Consider many alternative explanations for the observed effect

6. The Hill guidelines are useful in determining causation

7. Sensitivity analyses assesses model uncertainty and missing data

# 4 Covariation

1. Before choosing a measure of covariation, determine the source of the data, the nature of variables, and the symmetry status of the measure

2. Do not summarize regression sampling schemes with correlation

3. Do not correlate rates or ratios indiscriminately

4. To determine the appropriate sample size to estimate a population correlatiob $$\rho$$, use the following $$\Delta$$ in Rule 1 of sample size

$\Delta=\frac{1}{2}ln\frac{1+\rho}{1-\rho}$

1. Do not pair unless the correlation between the pairs is $$>$$ 0.5

2. Go beyond correlation in drawing conclusions, particularly in instances where location and scale are relevant

3. Assess agreement in terms of accuracy, scale differential, and precision

4. Assess test reliability by means of agreement

5. The range of the predictor variable determines the precision of the regression

6. In measuring change, width (i.e. spacing of the observations) is more important than the number of observations

# 5 Environmental Studies

1. Begin with the lognormal distribution in environmental studies

2. Differences are more symmetrical

3. Know the sample space for statements of risk

4. Beware of pseudo-replication (Hurlbert 1984)

5. Always consider alternatives to simple random sampling for a potential increase in efficiency, lower costs, and validity

6. In assessing the importance of an effect, consider the size of the population to which it applies

7. Models estimating small effects in large populations are particularly sensitive to assumptions. Extensive sensitivity studies are needed in such cases to validate the model

8. In assessing variation, distinguish between variability and uncertainty

9. In using a database, first look at the metadata, then look at the data

10. Always assess the statistical basis for an environmental standard

11. How a pollutant is measured plays a key role in identification, regulation, enforcement and remediation

12. Parametric analysis make maximum use of the data

13. Distinguish between confidence, prediction, and tolerance intervals (Vardeman 1992)

14. Risk assessment is divided into 5 areas - hazard identification, dose-response evaluation, exposure assessment, risk characterization, and risk management. Statistics plays an important role in the first 4. The last involves policy based on the first 4

15. Exposure and disease are usually widely separated both in space and time. Retrospective assessment of exposure is very difficult - particularly if the causes and mechanisms are poorly understood

16. Calibration involves inverse regression, and the error associated with the regression must be assessed

# 6 Epidemiology

1. Start with the poisson distribution to model disease incidence or prevalence

2. For a rare disease, the odds ratio approximates the relative risk

3. To detect a relative risk R in a rare disease cohort study, the number of exposed subjects (or unexposed subjects) $$n$$ for $$\alpha$$=0.05 and power = 0.8 is given by $n = \frac{4}{\pi_0(\sqrt{R-1})^2}$ where $$\pi_0$$ is the probability of the disease in the unexposed population and $$R$$ is the relative risk assumed to be >1

4. The estimate of sample size per group in a cohort study, based on the logarithm of the relative risk $$R$$ is given by $n= \frac{8(R+1)/R}{\pi_0(ln R)^2}$ for $$\alpha$$=0.05, power=0.8 and a two-sided alternative

5. Take no more than 4 to 5 controls per case

6. In logistic regression situations, about 10 events per variable are necessary inorder to get reasonably stable estimates of the regression coefficients

7. Begin with the exponential distribution to model time to event

8. Begin with two exponentials for comparing survival times

9. Be wary of surrogates. Accept substitutes warily

10. In rare diseases, the prevalence dominates the predictive value of a positive test

11. Do not dichotomize unless absolutely necessary

12. Select an additive or multiplicative model according to the following order: theoretical justification, practical implication, and computer implementation

# 7 Evidence Based Medicine

1. There are three hierarchies of evidence, each of which depend on the question asked and on the population of interest

2. The distinction between patient-oriented (POEM) and disease oriented (DOE) evidence is almost completely the difference between a surrogate endpoint and clinically relevant endpoint

3. In comparing two treatment regimens with binary outcome, start with absolute risk reduction

4. Number neeeded to treat (NNT) is a very useful clinical statistic but must be handled with care

5. Variability in treatment effect must always be considered over and above the average effect

6. Evidence for safety is limited

7. Intent to treat (ITT) is the default strategy for analysis

8. In EBM, it is more useful to discuss information about the prior rather than the prior

9. The four key questions for meta-analysis are the same as those in rule 1 of Basics

# 8 Design, Conduct, and Analysis

1. Randomization puts systematic sources of variability into the error term

2. Blocking is the key to reducing variability

3. Factorial design should be used to assess the joint effects of variables

4. Higher order effects occur rarely. Therefore it is not necessary to design experiments to incorporate higher order effects

5. Aim for balance in the design of a study

7. Assess independence, equal variance, and normality in that order

8. For every analysis, there is an appropriate graphical display

9. Distinguish between design structure and treatment structure of a study

10. Plan to do a hierarchical analysis of treatment effects by including all lower order effects associated with a higher order effect

11. Distinguish between nested and crossed design. The analysis will be quite different

12. Plan for missing data

13. Develop a strategy for dealing with multiple comparisons before starting a study

14. Know what properties a transformation preserves or does not preserve

15. Think of bootstrapping instead of the delta method in estimating complex relationships

# 9 References

## 9.1 Books

Agresti, Alan, An Introduction to Categorical Data Analysis, Wiley-Interscience, 2007.

Rosenbaum, Paul R. Observational Studies (second edition), Springer New York, 2002.

Cohen, Jacob Statistical Power Analysis for the Behavioral Sciences Routledge, 2nd edition, 1988

Cameron, Colin and Trivedi, Pravin Regression Analysis of Count Data Cambridge University Press, 2nd edition, 2013

## 9.2 Papers

Marcus-Roberts, Roberts, Meaningless Statistics, Journal of Educational Statistics, 1987.

Hill, A. B. The Environment or Disease: Association or Causation?, Presidential Address to the Section of Occupational Medicine of the Royal Society of Medicine, 1965.

Vardeman, S. B. What about the other intervals?, The American Statistician, Vol 46, No. 3, Aug 1992, pp 193-197

Hurlbert, S. H. Pseudoreplication and the design of ecological field experiments, Ecological Monographs, 1984, pp 187-211

## 9.3 Articles

Malinas, Gary and Bigelow, John, Simpson’s Paradox, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.)

Sandman, Peter, Mass Media and Environmental Risk: 7 Principles, 1997.