Biostatistics For Dummies
Overview
Break down biostatistics, make sense of complex concepts, and pass your class
If you're taking biostatistics, you may need or want a little extra assistance as you make your way through. Biostatistics For Dummies follows a typical biostatistics course at the college level, helping you understand even the most difficult concepts, so you can get the grade you need. Start at the beginning by learning how to read and understand mathematical equations and conduct clinical research. Then, use your knowledge to analyze and graph your data. This new edition includes more example problems with step-by-step walkthroughs on how to use statistical software to analyze large datasets. Biostatistics For Dummies is your go-to guide for making sense of it all.
- Review basic statistics and decode mathematical equations
- Learn how to analyze and graph data from clinical research studies
- Look for relationships with correlation and regression
- Use software to properly analyze large datasets
Anyone studying in clinical science, public health, pharmaceutical sciences, chemistry, and epidemiology-related fields will want this book to get through that biostatistics course.
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About The Author
Monika Wahi, MPH, CPH, leads the data science consulting firm DethWench Professional Services (DPS). She is also author of 8 LinkedIn Learning courses. John C. Pezzullo, PhD, held academic positions at Wayne State University and George-town University, including in the department of biomathematics and biostatistics.
Sample Chapters
biostatistics for dummies
CHEAT SHEET
To estimate sample size in biostatistics, you must state the effect size of importance, or the effect size worth knowing about. If the true effect size is less than the “important” size, you don’t care if the test comes out nonsignificant. With a few shortcuts, you can pick an important effect size and find out how many participants you need, based on that effect size, for several common statistical tests.
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Biostatistics, in its present form, is the cumulative result of four centuries of contributions from many mathematicians and scientists. Some are well known, and some are obscure; some are famous people you never would’ve suspected of being statisticians, and some are downright eccentric and unsavory characters.
Biology
The idea of a sampling distribution is at the heart of the concepts of accuracy and precision. Imagine a scenario in which an experiment (like a clinical trial or a survey) is carried out over and over again an enormous number of times, each time on a different random sample of subjects.
Using the "percent of kids who like chocolate" example, each experiment could consist of interviewing 50 randomly chosen children and reporting what percentage of kids in that sample said that they liked chocolate.
Biology
The four basic mathematical operations are addition, subtraction, multiplication, and division (ah, yes — the basics you learned in elementary school). Different symbols indicate these operations.
Addition and subtraction
Addition and subtraction are always indicated by the + and – symbols, respectively, placed between two numbers or variables.
To estimate sample size in biostatistics, you must state the effect size of importance, or the effect size worth knowing about. If the true effect size is less than the “important” size, you don’t care if the test comes out nonsignificant. With a few shortcuts, you can pick an important effect size and find out how many participants you need, based on that effect size, for several common statistical tests.
Biology
Over the years, as computing has moved from mainframes to minicomputers to personal computers to hand-held devices (calculators, tablets, and smartphones), statistical software has undergone a similar migration. Today you can find statistical software for just about every intelligent (that is, computerized) device there is (with the possible exception of smart toasters).
The basic idea of the median (that half of your numbers are less than the median) can be extended to other fractions besides 1/2. A centile is a value that a certain percentage of the values are less than. For example, 1/4 of the values are less than the 25th centile (and 3/4 of the values are greater). The median is just the 50th centile.
If the case report form (CRF) has been carefully and logically designed, entering each subject's data in the right place on the CRF should be straightforward. Then you need to get this data into a computer for analysis.
You can enter your data directly into the statistics software you plan to use for the majority of the analysis, or you can enter it into a general database program such as MS Access or a spreadsheet program like Excel.
Biology
Commercial statistical programs usually provide a wide range of capabilities, personal user support (such as a phone help-line), and some reason to believe (or at least to hope) that the software will be around and supported for many years to come.
Prices vary widely, and the array of pricing options may be bewildering, with single-user and site licenses, nonprofit and academic discounts, one-year and permanent licenses, "basic" and "pro" versions, and so on.
Biology
You may wonder why there are so many tests for such a simple task as comparing averages. Well, "comparing averages" doesn't refer to a single task; it's a broad term that can apply to a lot of situations that differ from each other on the basis of:
Whether you're looking at changes over time within one group of subjects or differences between groups of subjects (or both)
How many time points or groups of subjects you're comparing
Whether or not the numeric variable you're comparing is nearly normally distributed
Whether or not the numbers have the same spread (standard deviation) in all the groups you're comparing
Whether you want to compensate for the possible effects of some other variable on the variable you're comparing
These different conditions can occur in any and all combinations, so there are lots of possible situations.
In biostatistics, it's important to be comfortable with the basic concepts and terminology related to confidence intervals. This is an area where nuances of meaning can be tricky, and the right-sounding words can be used the wrong way.
Defining confidence intervals
Informally, a confidence interval indicates a range of values that's likely to encompass the true value.
Because you can't examine the entire population of people with the condition you're studying, you must select a representative sample from that population. You do this by explicitly defining the conditions that determine whether or not a subject is suitable to be in the study.
Inclusion criteria are used during the screening process to identify potential subjects and usually involve subject characteristics that define the population you want to draw conclusions about.
Biology
The proportion of subjects having some attribute (such as responding to treatment) can be compared between two groups of subjects by creating a cross-tab from the data, where the two rows represent the two groups, and the two columns represent the presence or absence of the attribute. In biostatistics, this cross-tab can be analyzed with a chi-square or Fisher Exact test.
Biology
For a correlation test in biostatistics (such as the Pearson or Spearman test), pick the scatter chart that looks like an important amount of correlation. Each chart shows the value of r (the correlation coefficient) and the required number of analyzable subjects (each providing an x and a y value). For example, if the scatter chart in the lower left corner (corresponding to r = 0.
Biology
While many programs, apps, and web pages are available to perform power and sample-size calculations, they aren’t always easy or intuitive to use. Because spreadsheets like Excel are readily available and intuitive, it’s convenient to have a single spreadsheet that can perform power and sample-size calculations for the situations that arise most frequently in biological and clinical research.
Most mathematical operators are written between the two numbers they operate on, or before the number if it operates on only one number (like the minus sign used as a unary operator). But factorials and absolute values are two mathematical operators that appear in typeset expressions in peculiar ways.
Factorials
Lots of statistical formulas contain exclamation points (!
Biology
Most of the approximate methods for determining confidence limits are based on the assumption that your sample statistic has a sampling distribution that's (at least approximately) normally distributed. Fortunately, there are good theoretical and practical reasons to believe that almost every sample statistic you're likely to encounter in practical work will have a nearly normal sampling distribution, for large enough samples.
Over the years, many dedicated and talented people have developed statistical software packages and made them freely available worldwide. Although some of these programs may not have the scope of coverage or the polish of the commercial packages, they're high-quality programs that can handle most, if not all, of what you probably need to do.
Biology
All statistical tests are derived on the basis of some assumptions about your data, and most of the classical significance tests (such as Student t tests, analysis of variance, and regression tests) assume that your data is distributed according to some classical frequency distribution (most commonly the normal distribution).
Biology
Two-dimensional arrays can be thought of as describing tables of values, with rows and columns (like a block of cells in a spreadsheet), and even higher-dimensional arrays can be thought of as describing a whole collection of tables.
Suppose you measure the fasting glucose on five subjects on each of three treatment days.
Biology
Setting up your data collection forms and database tables for categorical data requires more thought than you may expect. Everyone assumes he knows how to record and enter categorical data — you just type what that data is (for example, Male, White, Diabetes, or Headache), right? Bad assumption!
Carefully coding categories
The first issue is how to "code" the categories (how to represent them in the database).
The unpaired (independent-sample) t tests, one-way ANOVA, ANCOVA, and their nonparametric counterparts deal with comparisons between two or more groups of independent samples of data, such as different groups of subjects, where there's no logical connection between a specific subject in one group and a specific subject in another group.
Biology
Comparing within-group changes between groups is a special situation, but one that comes up very frequently in analyzing data from clinical trials. Suppose you're testing several arthritis drugs against a placebo, and your efficacy variable is the subject's reported pain level on a 0-to-10 scale. You want to know whether the drugs produce a greater improvement in pain level than the placebo.
Biology
Every research database, large or small, simple or complicated, should be accompanied by a data dictionary that describes the variables contained in the database. It will be invaluable if the person who created the database is no longer around. A data dictionary is, itself, a data file, containing one record for every variable in the database.
Biology
Most clinical trials have incomplete data for one or more variables, which can be a real headache when analyzing your data. The statistical aspects of missing data are quite complicated, so you should consult a statistician if you have more than just occasional, isolated missing values. Here are some commonly used approaches to coping with missing data:
Exclude a case from an analysis if any of the required variables for that analysis is missing.
Biology
Analytical populations are precisely defined subsets of the enrolled subjects that are used for different kinds of statistical analysis. Most clinical trials include the following types of analytical populations:
The safety population: This group usually consists of all subjects who received at least one dose of any study product (even a placebo) and had at least one subsequent safety-related visit or observation.
Biology
When you enter numerical data into your computer, don't combine two numbers into a single variable (such as 145/85 for systolic and diastolic blood pressure). When it comes to dates and times, however, exactly the opposite is true!
Most statistical software can represent dates and times as a single variable (an "instant" on a continuous timeline), so take advantage of that if you can — enter the date and time as one variable (for example, 07/15/2010 08:23), not as a date variable and a time variable.
Biology
Measurement accuracy very often becomes a matter of properly calibrating an instrument against known standards. The instrument may be as simple as a ruler or as complicated as a million-dollar analyzer, but the principles are the same. They generally involve the following steps:
Acquire one or more known standards from a reliable source.
Biology
Every time you perform a statistical significance test, you run a chance of being fooled by random fluctuations into thinking that some real effect is present in your data when, in fact, none exists.
This scenario is called a Type I error. When you say that you require p alpha level or saying that you want to limit your Type I error rate to 5 percent.
You improve the precision of anything you observe from your sample of subjects by having a larger sample. The central limit theorem (or CLT, one of the foundations of probability theory) describes how random fluctuations behave when a bunch of random variables are added (or averaged) together. Among many other things, the CLT describes how the precision of a sample statistic depends on the sample size.
Biology
An interim analysis is one that's carried out before the conclusion of a clinical trial, using only the data that has been obtained so far. Interim analyses can be blinded or unblinded and can be done for several reasons:
An institutional review board (IRB) may require an early look at the data to ensure that subjects aren't being exposed to an unacceptable level of risk.
Biology
A protocol is a document that lays out exactly what you plan to do in a clinical study. Ideally, every study involving human subjects should have a protocol.
Standard elements of a protocol
A formal drug trial protocol usually contains most of the following components:
Title: A title conveys as much information about the trial as you can fit into one sentence, including the protocol ID, study name (if it has one), clinical phase, type and structure of trial, type of randomization and blinding, name of the product, treatment regimen, intended effect, and the population being studied (what medical condition, in what group of people).
For numerical data, the main question is how much precision to record. Recording a numerical variable to as many decimal places as you have available is usually best.
For example, if a scale can measure body weight to the nearest 1/10 of a kilogram, record it in the database to that degree of precision. You can always round it off to the nearest kilogram later if you want, but you can never "unround" a number to recover digits you didn't record in the first place.
Probably the most general error-propagation technique is called Monte-Carlo analysis. You can use this technique to solve many difficult statistical problems. Calculating how SEs propagate through a formula for y as a function of x works like this:
Generate a random number from a normal distribution whose mean equals the value of x and whose standard deviation is the SE of x.
Biology
What do you do with the basic summary statistics that convey a general idea of how a set of numbers is distributed? Generally, when presenting your results, you pick a few of the most useful summary statistics and arrange them in a concise way. Many biostatistical reports select N, mean, SD, median, minimum, and maximum, and arrange them something like this:
mean ± SD (N)
median (minimum – maximum)
For example, the data (84, 84, 89, 91, 110, 114, and 116), the preceding arrangement looks like this:
98.
A categorical variable is summarized in a fairly straightforward way. You just tally the number of subjects in each category and express this number as a count — and perhaps also as a percentage of the total number of subjects in all categories combined. So, for example, a sample of 422 subjects can be summarized by race.
You can run the Student t tests using typical statistical software and interpret the output produced. In this example, you'll be using the software package OpenStat.
The basic idea of a t test
All the Student t tests for comparing sets of numbers are trying to answer the same question, "Is the observed difference larger than what you would expect from random fluctuations alone?
Biology
The first step (Phase I) in human drug testing is to determine how much drug you can safely give to a person, which scientists express in more-precisely defined terms:
Dose-limiting toxicities (DLTs) are unacceptable side effects that would force the treatment to stop (or continue at a reduced dose). The term unacceptable is relative; severe nausea and vomiting would probably be considered unacceptable (and therefore DLTs) for a headache remedy, but not for a chemotherapy drug.
Biology
After the Phase I trials of human drug testing, you'll have a good estimate of the maximum tolerated dose (MTD) for the drug. The next step is to find out about the drug's safety and efficacy at various doses. You may also be looking at several different dosing regimens, including the following options:
What route (oral or intravenous, for example) to give the drug
How frequently to give the drug
For how long (or for what duration) to give the drug
Generally, you have several Phase II studies, with each study testing the drug at several different dose levels up to the MTD to find the dose that offers the best tradeoff between safety and efficacy.
Biology
If Phase II of human drug testing is successful, it means you've found one or two doses for which the drug appears to be safe and effective. Now you take those doses into the final stage of drug testing: Phase III.
Phase III is kind of like the drug's final exam time. It has to put up or shut up, sink or swim, pass or fail.
Biology
Being able to market the drug doesn't mean you're out of the woods yet! During a drug's development, you've probably given the drug to hundreds or thousands of subjects, and no serious safety concerns have been raised. But if 1,000 subjects have taken the drug without a single catastrophic adverse event, then that only means that the rate of these kinds of events is probably less than 1 in 1,000.
Biology
The aims or goals of a study are short general statements (often just one statement) of the overall purpose of the trial. For example, the aim of a study may be "to assess the safety and efficacy of drug XYZ in patients with moderate hyperlipidemia."
The objectives are much more specific than the aims. Objectives usually refer to the effect of the product on specific safety and efficacy variables, at specific points in time, in specific groups of subjects.
Around the middle of the 20th century, the idea of levels of measurement caught the attention of biological and social-science researchers, and, in particular, psychologists. One classification scheme, which has become very widely used (at least in statistics textbooks), recognizes four different levels at which variables can be measured: nominal, ordinal, interval, and ratio:
Nominal variables are expressed as mutually exclusive categories, like gender (male or female), race (white, black, Asian, and so forth), and type of bacteria (such as coccus, bacillus, rickettsia, mycoplasma, or spirillum), where the sequence in which you list a variable's different categories is purely arbitrary.
One of the reasons (but not the only reason) for running a multiple regression analysis is to come up with a prediction formula for some outcome variable, based on a set of available predictor variables. Ideally, you’d like this formula to be parsimonious — to have as few variables as possible, but still make good predictions.
Biology
Before any proposed treatment can be tested on humans, there must be at least some reason to believe that the treatment might work and that it won't put the subjects at undue risk. So every promising chemical compound or biological agent must undergo a series of tests to assemble this body of evidence before ever being given to a human subject.
Biology
Several other kinds of means, besides arithmetic, are useful measures of central tendency in certain circumstances. They're called means because they all involve the same "add them up and divide by how many" process as the arithmetic mean, but each one introduces a slightly different twist to the basic process.
Biology
As you dive deeper into the field of biostatistics, you'll need to develop a firm understanding of pharmacokinetics (PK) and pharmacodynamics (PD) and the differences between the two.
The term pharmacokinetics (PK) refers to the study of
How fast and how completely the drug is absorbed into the body (from the stomach and intestines if it's an oral drug)
How the drug becomes distributed through the various body tissues and fluids, called body compartments (blood, muscle, fatty tissue, cerebrospinal fluid, and so on)
To what extent (if any) the drug is metabolized (chemically modified) by enzymes produced in the liver and other organs
How rapidly the drug is eliminated from the body (usually via urine, feces, and other routes)
The term pharmacodynamics (PD) refers to the study of
The relationship between the concentration of the drug in the body and the biological and physiological effects of the drug on the body or on other organisms (bacteria, parasites, and so forth) on or in the body.
The idea of sampling from a population is one of the most fundamental concepts in statistics — indeed, in all of science. For example, you can't test how a chemotherapy drug will work in all people with lung cancer; you can study only a limited sample of lung cancer patients who are available to you and draw conclusions from that sample.
Biology
These three mathematical operations — working with powers, roots, and logarithms — are all related to the idea of repeated multiplication. These basic functions are used to help build more complex formulas.
Raising to a power
Raising to a power is a shorthand way to indicate repeated multiplication. You indicate raising to a power by
Superscripting in typographic formulas, such as 53 = 125
** in plain text formulas, such as 5**3 = 125
^ in plain text formulas, such as 5^3 = 125
All the preceding expressions are read as "five to the third power," or "five cubed," and tell you to multiply three fives together: 5 × 5 × 5, which gives you 125.
Samples differ from populations because of random fluctuations. Statisticians understand quantitatively how random fluctuations behave by developing mathematical equations, called probability distribution functions, that describe how likely it is that random fluctuations will exceed any given magnitude.
A probability distribution can be represented in several ways:
As a mathematical equation that gives the chance that a fluctuation will be of a certain magnitude.
After you've designed your study and have described it in the protocol document, it's time to set things in motion. In any research involving human subjects, two issues are of utmost importance:
Safety: Minimizing the risk of physical harm to the subjects from the product being tested and from the procedures involved in the study
Privacy/confidentiality: Ensuring that data collected during the study is not made public in a way that identifies a specific subject without the subject's consent
Regulatory agencies
In the United States, several government organizations oversee human subjects' protection:
Commercial pharmaceutical research is governed by the Food and Drug Administration (FDA).
Biology
The word random is something folks use all the timeYou probably have some intuitive concept of randomness, but find may hard it to put into precise language. Random is a term that applies to the data you acquire in your experiments. You can talk about random events and random variables.
When talking about a sequence of numbers, random means the absence of any pattern in the numbers that could be used to predict what the next number will be.
Biology
In biostatistics, when comparing paired measurements (such as changes between two time points for the same subject) using a paired Student t test, the effect size is expressed as the ratio of Δ (delta, the mean change) divided by σ (sigma, the standard deviation of the changes). Another, perhaps easier, way to express the effect size is by the relative number of expected subjects with positive versus negative changes.
Biology
In biostatistics, when comparing the means of two independent groups of subjects using an unpaired Student t test, the effect size is expressed as the ratio of Δ (delta, the difference between the means of two groups) divided by σ (sigma, the within-group standard deviation).
Each chart in the following figure shows overlapping bell curves that indicate the amount of separation between two groups, along with the effect size (Δ/σ) and the required number of analyzable subjects in each group.
Scientists conduct experiments on limited samples of subjects in order to draw conclusions that (they hope) are valid for a large population of people. Suppose you want to conduct an experiment to determine some quantity of interest. For example, you may have a scientific interest in one of these questions:
What is the average fasting blood glucose concentration in adults with diabetes?
Histograms are bar charts that show what fraction of the subjects have values falling within specified intervals. The main purpose of a histogram is to show you how the values of a numerical value are distributed. This distribution is an approximation of the true population frequency distribution for that variable.
Biology
Even though some general error-propagation formulas are very complicated, the rules for propagating SEs through some simple mathematical expressions are much easier to work with. Here are some of the most common simple rules.
All the rules that involve two or more variables assume that those variables have been measured independently; they shouldn't be applied when the two variables have been calculated from the same raw data.
Biology
Simple expressions (also called formulas) have one or two numbers and only one mathematical operator (for example, 5 + 3). But most of the formulas you'll encounter in biostatistics are more complicated, with two or more operators.
You need to know the order in which to do calculations, because using different sequences of operations produces different results.
Biology
The standard deviation (usually abbreviated SD, sd, or just s) of a bunch of numbers tells you how much the individual numbers tend to differ (in either direction) from the mean. It's calculated as follows:
This formula is saying that you calculate the standard deviation of a set of N numbers (Xi) by subtracting the mean from each value to get the deviation (di) of each value from the mean, squaring each of these deviations, adding up the
terms, dividing by N – 1, and then taking the square root.
Statistical decision theory is perhaps the largest branch of statistics. It encompasses all the famous (and many not-so-famous) significance tests — Student t tests, chi-square tests, analysis of variance (ANOVA;), Pearson correlation tests, Wilcoxon and Mann-Whitney tests, and on and on.
In its most basic form, statistical decision theory deals with determining whether or not some real effect is present in your data.
Statistical estimation theory focuses on the accuracy and precision of things that you estimate, measure, count, or calculate. It gives you ways to indicate how precise your measurements are and to calculate the range that's likely to include the true value.
Accuracy and precision
Whenever you estimate or measure anything, your estimated or measured value can differ from the truth in two ways — it can be inaccurate, imprecise, or both.
Biology
Sometimes you want to show how a variable varies from one group of subjects to another. For example, blood levels of some enzymes vary among the different races. Two types of graphs are commonly used for this purpose: bar charts and box-and-whiskers plots.
Bar charts
One simple way to display and compare the means of several groups of data is with a bar chart, like the one shown, where the bar height for each race equals the mean (or median, or geometric mean) value of the enzyme level for that race.
All the famous statistical significance tests (Student t, chi-square, ANOVA, and so on) work on the same general principle — they evaluate the size of apparent effect you see in your data against the size of the random fluctuations present in your data. Following are the general steps that underlie all the common statistical tests of significance.
Biology
The so-called “one-way analysis of variance” (ANOVA) is used when comparing three or more groups of numbers. When comparing only two groups (A and B), you test the difference (A – B) between the two groups with a Student t test. So when comparing three groups (A, B, and C) it’s natural to think of testing each of the three possible two-group comparisons (A – B, A – C, and B – C) with a t test.
Biology
You can calculate the standard error (SE) and confidence interval (CI) of the more common sample statistics (means, proportions, event counts and rates, and regression coefficients). But an SE and CI exist (theoretically, at least) for any number you could possibly wring from your data — medians, centiles, correlation coefficients, and other quantities that might involve complicated calculations, like the area under a concentration-versus-time curve (AUC) or the estimated five-year survival probability derived from a survival analysis.
Biology
No matter how they're written, mathematical formulas are just concise "recipes" that tell you how to calculate something or how something is defined. You just have to know how to read the recipe. To start, look at the building blocks from which formulas are constructed: constants (whose values never change) and variables (names that stand for quantities that can take on different values at different times).
A less extreme form of the old saying "garbage in equals garbage out" is "fuzzy in equals fuzzy out." Random fluctuations in one or more measured variables produce random fluctuations in anything you calculate from those variables. This process is called the propagation of errors. You need to know how measurement errors propagate through a calculation that you perform on a measured quantity.
Just as the SE (standard error) formulas depend on what kind of sample statistic you're dealing with (whether you're measuring or counting something or getting it from a regression program or from some other calculation), confidence intervals (CIs) are calculated in different ways depending on how you obtain the sample statistic.
If you were to survey 100 typical children and find that 70 of them like chocolate, you'd estimate that 70 percent of children like chocolate. What is the 95 percent confidence interval (CI) around that 70 percent estimate?
There are many approximate formulas for confidence intervals around an observed proportion (also called binomial confidence intervals).
Biology
This is one time you don't need any formulas because you shouldn't attempt to calculate standard errors or confidence intervals (CIs) for regression coefficients yourself. Any good regression program can provide the SE for every parameter (coefficient) it fits to your data. The regression program may also provide the confidence limits for any confidence level you specify, but if it doesn't, you can easily calculate the confidence limits using the formulas for large samples.
Biology
There are many approximate formulas for the CIs (confidence intervals) around an observed event count or rate (also called a Poisson CI). Suppose that there were 36 fatal highway accidents in your county in the last three months.
If that's the only safety data you have to go on, then your best estimate of the monthly fatal accident rate is simply the observed count (N), divided by the length of time (T) during which the N counts were observed: 36/3, or 12.
The theory of statistical hypothesis testing was developed in the early 20th century and has been the mainstay of practical statistics ever since. It was designed to apply the scientific method to situations involving data with random fluctuations (and almost all real-world data has random fluctuations). Following are a few terms commonly used in hypothesis testing.
The end result of a statistical significance test is a p value, which represents the probability that random fluctuations alone could have generated results that differed from the null hypothesis (H0), in the direction of the alternate hypothesis (HAlt), by at least as much as what you observed in your data.
If this probability is too small, then H0 can no longer explain your results, and you're justified in rejecting it and accepting HAlt, which says that some real effect is present.
The power of a statistical test is the chance that it will come out statistically significant when it should — that is, when the alternative hypothesis is really true. Power is a probability and is very often expressed as a percentage. Beta is the chance of getting a nonsignificant result when the alternative hypothesis is true, so you see that power and beta are related mathematically: Power = 1 – beta.
The range of a set of values in your data is the difference between the smallest value (the minimum value) and the largest value (the maximum value):
Range = maximum value – minimum value
So for the IQ example in the preceding section (84, 84, 89, 91, 110, 114, and 116), the minimum value is 84, the maximum value is 116, and the range is 32 (equal to 116 – 84).
Biology
You can use confidence intervals (CIs) as an alternative to some of the usual significance tests. To assess significance using CIs, you first define a number that measures the amount of effect you're testing for. This effect size can be the difference between two means or two proportions, the ratio of two means, an odds ratio, a relative risk ratio, or a hazard ratio, among others.
Biology
Biostatistics can be surprising sometimes: Data obtained in biological studies can often be distributed in strange ways, as you can see in the following frequency distributions:
Two summary statistical measures, skewness and kurtosis, typically are used to describe certain aspects of the symmetry and shape of the distribution of numbers in your statistical data.
Biology
In the mid-1900s it was recognized that certain drugs interfered with the ability of the heart to "recharge" its muscles between beats, which could lead to a particularly life-threatening form of cardiac arrhythmia called Torsades de Points (TdP). Fortunately, warning signs of this arrhythmia show up as a distinctive pattern on an electrocardiogram (ECG) well before it progresses to TdP.
Biology
Modern statistical software makes it easy for you to analyze your data in most of the situations that you’re likely to encounter (summarize and graph your data, calculate confidence intervals, run common significance tests, do regression analysis, and so on). But occasionally you may run into a problem for which no preprogrammed solution exists.
Two quite different ideas about probability have coexisted for more than a century. These probability approaches, which differ in several important ways, are as follows:
The frequentist view defines probability of some event in terms of the relative frequency with which the event tends to occur.
The Bayesian view defines probability in more subjective terms — as a measure of the strength of your belief regarding the true situation.
Biology
The outcome of a statistical test is a decision to either accept or reject H0 (the Null Hypothesis) in favor of HAlt (the Alternate Hypothesis). Because H0 pertains to the population, it's either true or false for the population you're sampling from. You may never know what that truth is, but an objective truth is out there nonetheless.
Biology
Statpages calculates how precision propagates through almost any expression involving one or two variables. It even handles the case of two variables with correlated fluctuations. You simply enter the following items:
The expression, using a fairly standard algebraic syntax (JavaScript)
The values of the variable or variables
The corresponding SEs
Consider the example of estimating the SE of the area of a circle whose diameter is 2.
Biology
It's often convenient, when dealing with collections of numbers, to use a single variable name to refer to the entire set of numbers. A bunch of values referred to by a single variable name is generally called an array. Arrays can have one or more dimensions, which you can think of as rows, columns, and slices.
Randomized controlled trials (RCTs) are the gold standard for clinical research. In an RCT, the subjects are randomly allocated into treatment groups (in a parallel trial) or into treatment-sequence groups (in a crossover design).
Randomization provides several advantages:
It tends to eliminate selection bias — preferentially giving certain treatments to certain subjects (assigning a placebo to the less “likeable” subjects) — and confounding, where the treatment groups differ with respect to some characteristic that influences the outcome.
A bioequivalence study is usually a fairly simple pharmacokinetic study, having either a parallel or a crossover design. You may be making a generic drug to compete with a brand-name drug already on the market whose patent has expired. The generic and brand-name drug are the exact same chemical, so it may not seem reasonable to have to go through the entire drug development process for a generic drug.
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