Introduction to Biostatistics with Chi-Squared Analysis

Probability is one tool we use in statistics, for quantifying uncertainty because we have imperfect data, and

modeling random events. Probability is applying the rules of probability to assign a comparative value to the

chance of events occurring. We use probability in statistics as a reference of what would happen by chance if

there were no associations occurring. Secondly, we use probability because we are modelling the probability of

what is happening not by chance but when there is an association.

Probability theory is grounded in three fundamental axioms that establish the rules governing the behavior of

probabilities. These axioms assign a comparative value to the chance of events occurring.

1. Non-Negativity Axiom:

The first axiom dictates that probabilities are non-negative. In mathematical terms, for any event A, the

probability P(A) is greater than or equal to zero. This fundamental constraint aligns with our intuitive

understanding that the likelihood of an event cannot be negative.

2. Additivity Axiom:

The second axiom pertains to the addition of probabilities. For any mutually exclusive events—events that

cannot occur simultaneously—the probability of their union is the sum of their individual probabilities. This

axiom provides a systematic way to combine probabilities when events are mutually exclusive.

3. Unitarity Axiom:

The third axiom, often referred to as the unitarity axiom, asserts that the sum of the probabilities of all possible

outcomes within a sample space is equal to 1. This axiom reflects the exhaustive nature of probabilities within a

given context, emphasizing that the certainty of an event occurring or not occurring is absolute.

Addition Rule

The addition rule manifests in two distinct forms to accommodate different scenarios involving the combination

of probabilities.

Mutually Exclusive Events:

For mutually exclusive events—events that cannot occur simultaneously—the addition rule stipulates that the

probability of their union is the sum of their individual probabilities: P(A or B) = P(A) + P(B). This straightforward

formula accounts for the absence of overlap between events.

Non-Mutually Exclusive Events:

However, in situations where events are not mutually exclusive and may have shared elements, the addition

rule requires a nuanced approach. To avoid double-counting shared probabilities, the formula becomes: P(A or

B) = P(A) + P(B) - P(A and B). This adjustment ensures that the shared probability is included only once in the

overall calculation.

Conditional probability is a probability in which the possibility of an event depends upon the occurrence of a previous event. Conditional probability is the likelihood of an outcome occurring based on a previous outcome in similar circumstances. You write it as A given B, expressed as P(A|B), indicating that the probability of event A is dependent on the occurrence of event B.

Independent Events:

When events are independent, the multiplication rule simplifies the calculation of the combined probability. The

probability of both events A and B occurring is the product of their individual probabilities: P(A and B) = P(A) *

P(B). This rule reflects the idea that the joint probability of independent events is the result of their individual

probabilities multiplied together.

Bayes' rule is being popularized in probabilistic reasoning, offering a dynamic framework for updating

probabilities based on new information or evidence. This rule is particularly valuable in situations where

probabilities need adjustment as additional data becomes available.

Components of Bayes' Rule:

Bayes' rule incorporates the following components:

- Prior Probability (P(A)): The initial probability or belief regarding the occurrence of an event before new

evidence is considered.

- Likelihood Function (P(B|A)): The probability of observing the new evidence given the occurrence of the

event.

- Evidence (P(B)): The overall probability of observing the evidence.

- Posterior Probability (P(A|B)): The updated probability of the event occurring after considering the new

evidence.

Mathematical Formulation:

Bayes' rule is expressed as:

P(A|B) = {P(B|A) / (P(A)*{P(B))

Applications:

Bayes' rule finds applications in diverse fields, from medical diagnosis and finance to artificial intelligence and

machine learning. It is particularly instrumental in scenarios where decision-making relies on incorporating

evolving information and adjusting probabilities accordingly.

Probability distributions are simply frequency distribution of some measurement that has had probability assigned to the outcomes. In this case, we will discuss categorical and normal distributions. Categorical Probability Distributions:A categorical probability distribution, also known as discrete probability distribution, involves random variables that possess outcomes that fall into distinct categories such as 'yes' or 'no', 'pass' or 'fail', 'heads' or 'tails', etc. General ly, each category has a corresponding probability, and the sum of all probabilities is equal to one.

Normal Distribution

On the other hand, the normal (or Gaussian) distribution is a continuous bell-shaped curve that is defined by its mean (µ=mu) and standard deviation (σ =sigma). It basically represents the measurement frequency of a continuous random process, and because of the nature of continuous variables, you can’t find the probability of one value, so the probability you estimate is the cumulative probability to that point usually coming from the left side of the number line.

The empirical rule, also known as the 68-95-99.7 rule, provides the estimates for the proportion of data lying within one, two, or three standard deviations of the mean in a normal distribution. It denotes that approximately 68%, 95%, and 99.7% of the observations fall within one, two, and three standard deviations from the mean, respectively. The cumulative probabilities can be found in a table and the area under the curve that it estimates is usually pictured at the top.

There are other rules you can learn also about distributions and the two most important ones besides the emperical rule that you should look at is the Law or Large Numbers, and the Central Limit Theory. Be sure to look those up.

As we know normal distribution assumes two important characteristics about the dataset: a large sample size and knowledge of the population standard deviation. However, if we do not meet these two criteria, and we have a small sample size or an unknown population standard deviation, then we use the t-distribution. Student's t-distribution, also known as Student's t-distribution, is a type of sampling distribution that is symmetric and bell-shaped, like the normal distribution, but has heavier tails.

One thing that is different than the normal distribution is the t-distributions curve shapes depend on the samples degrees of freedom. As the degrees of freedom increases, the t-distribution becomes more like a standard normal distribution.

Confidence Intervals

Confidence intervals help assess the reliability and error of estimates and make informed statistical inferences. When a sample is taken and the mean and standard deviation of the population of interest are unknown, it is practical cases, we use Confidence Intervals (CI) to show the range of the measurement and the local error within the sample, the sampling error. Different samples of the same population will give different results. This is called sampling error or variation due to sampling. The CI gives an estimated range of values likely to include an unknown population parameter - the way it works entails a desired level of confidence, usually expressed as a percentage (commonly 95%); therefore, it is a range estimator not a point estimator. A 95% CI states that 95% of the interval estimates will include the population mean. Let me clarify the population representation interpretation should not be taken too far from the characteristics of the sample because the sample you take and the error it represents is usually a local population and not more than local.

The width of a confidence interval depends on two things: The variation within the population of interest, and the size of the sample. If all the values in the population were mostly the same, then our sample will also have very little variation. Any sample we take is likely to be pretty similar to any other sample. Our estimate is going to be close to the true population value. We would have a small confidence interval. But a more varied population will lead to a more varied sample. Different samples taken of the same population will differ more. We would be less sure that the sample mean was close to the population mean. Our confidence interval would be larger. So, greater variation in the population leads to a wider confidence interval. Sample size also affects the width of a confidence interval. If we take a small sample, we don't have much information on which to base our inference. Small samples will vary more from each other. There is more variation due to sampling, or sampling error, and mean estimation with a small sample. In larger samples, the effect of a few unusual values is evened out by the other values in the sample. Larger samples will be more similar to each other. The effect of sampling error is reduced with larger samples. When we take a large sample, We have more information and can be more sure about our estimate.

The confidence interval can be smaller and that indicates less error which is more desirable as an estimator than one with more error. When we use traditional confidence interval formulas, the stated level of confidence also effects the width of the confidence interval. All estimates of population parameters, such as means, medians, differences of means and differences in medians should be expressed as confidence intervals. Probability the confidence level of 95 percent or 99 is used very often. It is basically using the Empirical Rule and you are stating this samples probable range of values.

If a confidence the set value for the respective confidence interval n is the sample size and s is the standard deviation plus minus s divided by the root of n where we take the minus for the lower bound and we have the upper limit with x dash plus for the confidence level of 95 for example the set value is 1.96 times the standard deviation divided by the root of n. You could instead repeated sample from a population 100 or 1000 or etc., times and create a confidence interval that way and that’s called bootstrapping. I dont think it is considered that the true value of the population parameter lies in the interval. Formula is usually and here for the Z estimator

Xbar±Zalpha=0.95×s/sqrt(n)

where Xbar is the sample mean, s is the sample standard deviation, n is the sample size, and Z is the z-score corresponding to the desired confidence level.

Here's an Example problem

A study on cholesterol levels in patients with diabetes reveals that the parameter is normally distributed with a mean of 230 mg/dL and standard deviation of 10 mg/dL. Using that information, calculate the 95% confidence interval which would represent 95% of serum cholesterol observations in these patients?

a. 200-260 mg/dL

b. 210-250 mg/dL

c. 225-235 mg/dL

d. 220-240 mg/dL

e. 220-260 mg/dL

The Binomial distribution is a sampling probability distribution used a lot in Biostatistics, it is basically used to model the number of successes in a number of independent trials, where the outcome of the trial has only two possible outcomes. The Binomial distribution is used to model rates.

The Binomial as with any distribution can be used to calculate the probability of getting exactly one number. I cover that here and how the binomial probability function works.

The Binomial as with any distribution can be used to calculate various probabilities. For example, the probability of getting that number or less, etc. I show a online calculator and how it shows the other probabilities cthat can be calculates using the binomial distribution.

The Kaplan Meier method is a method of calculating the probability of events and survival over time.  The reason I put this method here is because the method uses probability only to summarize the events.

The Kaplan Meier can also be plotted that is part of the Kaplan Meier method. Again you are graphing the probability of survival over time. I will attach a sample size calculation for a single survival curve and for comparing two curves using the Log Rank test.

The Kaplan Meier can also be plotted that is part of the Kaplan Meier method. Again you are graphing the probability of survival over time. Im also giving you a short description of the log rank test for which I gave you the sample size for in the previous video.

Diagnostic tests aren't perfect. Because of that, we have measures that describe how well the diagnostic test does two jobs: 1 is how well does the test identify the sick participants, 2 is how well does the test identify the not sick participants.

Because a diagnostic test is not perfect, we use a confusion table to describe the measures that summarize the adequacy of the test. It uses both the test being evaluated and the reference which is considered to be the closest to the truth as possible. That test which is the best at the time is called the Gold Standard.

How can we use probabilities to calculate the probability a diagnostic test has accurately identified positive participants and negative participants. We use the Bayes Rule. I showed you in the probability section the Bayes Rule and how it works. We use an HIV rapid test as our example.

I show using the Bayes probability formula, to determine the probability that a participant is actually sick when the rapid test is positive. It is a big formula but I show exactly where you can get the information for all the terms.

I continue define the terms in the Bayes Rule Formula using an HIV rapid test. I show which are the false positives and true positives probabilities and the rates describing those. Also show how a clinical trial is needed to validate a diagnostic test.

This is the confusion table used to show all the estimators used to evaluate the new diagnostic test. There is sensitivity, who well the test identifies true sick participants (true positive). There is specificity is the probability that the participant that is really negative (true negative). We show the testing algorithm.

Randomization is none of the most important parts of clinical research. Randomization has been found to be a vital part of causal analysis. We won't be doing causal analysis here. Randomization is used to rule out bias when you are assigning experimental units to a group. When you standardize units to treatment and control you are removing any treatment assignment bias and, in theory, when you randomize the characteristics of the samples should be represented equally in the two groups. I show you how to randomize in R. I first discuss in this video that the block sizes are randomized to reduce bias in treatment assignment that may occur if the pattern of randomization was deciphered by recruiters, interviewers or interventionist.

I show you how to randomize in R. I show you which package I suggest you use to randomize and how to specify the block sizes. And how R uses the function to create a randomization list.

I show you how to randomize in R. I go over the output of the R package we used to create the randomization list.

Prevalence of a Disease:

The prevalence of a disease is a measure of how widespread the disease is within a population at a

particular point in time. It's calculated by dividing the number of existing cases of the disease by the total

population at risk and multiplying by 100 to express it as a percentage.

Prevalence = (Number of existing cases / Total population at risk) * 100

For example, let's consider a population of 10,000 people living in a city. Among them, 500 individuals

have been diagnosed with hypertension. The prevalence of hypertension in this population would be:

Prevalence = (500 / 10,000) * 100 = 5%

This means that 5% of the population currently has hypertension.

Incidence of a Disease:

While prevalence tells us how many people have a disease at a specific point in time, incidence measures

the rate at which new cases of the disease develop within a population over a defined period. It's

calculated by dividing the number of new cases of the disease by the total population at risk and

multiplying by 100 to express it as a percentage.

Incidence Rate = (Number of new cases / Total population at risk) * 100

For instance, if we observe 50 new cases of diabetes in the same population of 10,000 people over one

year, the incidence rate of diabetes would be:

Incidence Rate = (50 / 10,000) * 100 = 0.5%

This indicates that 0.5% of the population developed diabetes within that one-year period

Incidence and prevalence studies are usually done with two different study designs. The incidence study is used to identify wen new cases occur and how often you usually would use a cohort study. A cohort study is determining a group of people (cohort) that you are interested in following forward in time to determine the groups incidence rate.

The study usually used to determine the prevalence of a disease in a population is a cross-sectional study using just a survey at one point in time to ask are the persons sick: now or have they been in the past for a certain length of time.

Incidence and prevalence rates can be calculated in various ways and I go over the crude rate, the age adjusted rate, the gender adjusted rate and the age and gender adjusted rate.

I show a graph to go over the crude rate, the age adjusted rate, the gender adjusted rate and the age and gender adjusted rate. Remember, in research articles, not to put the graph and the words because it is like repeating yourself. Always check the instructions for authors for the individual journal you are planning to publish in.

When you are reading the literature read the fine print so you can understand exactly what is being shown. Also be sure to ask whomever is asking for a rate you ask them to specify exactly what rate they want and if they want it adjusted for anything. That will save you some time.

Is this video I go over convenience sampling random sampling and systematic sampling. I also show how it could be done in the field and what are the measures you should measure to make sure you can describe in the article what the sampling characteristic are. We go over the sampling fraction.

We start reviewing the 7 steps of the Hypothesis Testing Framework and the different criteria setup to complete a hypothesis test. We begin with defining the Null and Alternative hypotheses. We show the test-statistic, which is the heart of statistics, and show that each test has a different test-statistic.

We begin with the confusion matrix that is used for the 4 outcomes of a Hypothesis Test. We are using the table to show the characteristic of each value of reality and each value of the decision from the hypothesis test. We go over which each cell of the table represents and how we assign probability to the table outcomes.  Show the type 1 error.

We continue to discuss the confusion matrix of a Hypothesis Test. We are using the table to show the characteristic of each value of reality and each value of the decision from the hypothesis test. We go over which each cell of the table represents and how we assign probability to the table outcomes.  We show the type 2 error.

We continue to the step 3 of the Hypothesis Test Framework. We introduce sample size and power and we begin an example of the hypothesis testing of the t-test. We setup the test-statistics for the two sample t-test.

We continue the final steps of the Hypothesis Testing Framework for the two-sample t-test. WEe discuss the alpha and beta and two sided test requirements for the hypothesis testing. We also discuss the final step of comparing the test statistic value you calculate from your study data, realize we took your whole study and turned it into 1 number. This is a discussion happening now that that might not be the best way to analyze clinical trials.

We begin the paired t-test. We introduce an example which was first clinical trial analysis. We talked about why we use paired t-test. We introduce a idea about the variance in difference between the same persons eye is lower than the variance of the difference between 2 peoples eyes. This will we be investigated further in another course called Design of Experiments.

We begin to fill out the hypothesis testing framework for the paired t-test. We introduce an example which was first clinical trial analysis. We talked about why we use paired t-test and how the test statistic is calculated. I probably shouldn't have used y as one of my variable names, it probably would have been better to use the x of A and X of b for the difference variables but as long as you realize how the difference population is calculated then it's all good.

We continue fill out the hypothesis testing framework for the paired t-test and how the test statistic is calculated.

We finish off the paired t-test. We begin to show the test statistic is based on the difference population and give the assumptions needed to run a paired t-test hypothesis test.

I decided to move sample size below the first sample size calculation video. It may contain some information that is for future videos such as the testing of two proportions. What I suggest you do is: watch the videos on sample size and bias and if you prefer to watch them again after the chi-square analysis has been discussed, I think it would be a good idea to watch it again. Note: I used an avatar to make this video. See what you think. I find it to have a very flat affect.

We begin discussing the doctors or the study expert's important role in determining the sample size. This is where medicine and statistics meet. The doctor have to decide what is the minimum amount of a difference in the outcome that can be measured and where there is a identifiable different in a patients health condition.

We discuss the power of a statistical test. The ability to see a difference when there is a difference is a shorthand definition. We also discuss only a few Biases and I give a link to a website that has a wole library of research biases students can look at.

We start with the sample size calculation. We show a fancy sample size calculator software might not be needed at this level of the statistical tests we are preforming. We then discuss the different pieces of information you have to bring with you to fill out the sample size calculators. We discuss in the sample size class that you need at least 3 pieces of information

We explore the different values and their effect on the sample size using an online sample size calculator.

Finally, we perform a sample size adequate to ensure a statistically significant finding given the study parameters entered into it. Remember that the sample size is only as accurate as the values you put in it. I also give a small set of slides on p-values and how to interpret them. There is a discussion also on what a p-vlaue is and how to interpret them you may want to research.

The Chi-squared test is test of homogeneity of categorical groups. We begin be discussing the one column chi-squared which is used to determine if the categories of a single variable are equally distributed. We discuss the hypotheses of the one column chi-squared test and the test statistics. We begin discussing the chi-squared

We continue to show the outcomes of an example and the actual calculations.We discuss the formula and the differences between a dichotomous variable test and a multi categorical variable test.  We also discuss how the degrees of freedom are used in the test.

We finalize the calculation of the single column calculation. and its p-value. We the move to the two categorical variable tests using the chi-squared analysis. We begin with the 2x2 tables and the first sampling condition which is analyzed using the test of association or if the 2 variables are independent and have no association.

We introduce the z test formulas for the independence test of association between 2 categorical measures. We also discuss the labeling of the 2x2 table for use in the statistical formulas. There is a table of ns representing the numbers that are in the table and there is a table of ps which represent the proportions in each table.

We discuss the studies associated with different uses of the 2x2 table. We discuss retrospective studies and discuss that the chi-squared test can be used for all three of the 2x2 tables that are presented and the sampling designs. Fisher's Exact test and not using the chi-squared for longitudinal analysis.

Begin calculating a 2x2 chi-squared test. We discuss the different specification for the estimators: the row and column proportions. And we discuss the prevalence table.

We begin the chi-squared chi-square test statistic and discuss how the expected values of the chi-square are calculated.

We show a small explanation of how the expected calculation works.

We use the Chi-Squared formula and our expected value in the test statistic. We show an important feature of the table and that is as the sample size rises the p-value gets smaller. this is an important idea. You have control over what value the p-vale has by setting the sample size. We will discuss this further in the sample size videos.

Here we talk about the different estimators and measures that can be used to analyze and report different proportions such as the risk ratio, etc.

We start using SPSS to analyze a 2x2 crosstab table. We calculate the same chi-square estimate and the p-value in SPSS that we calculated by hand.

We show the same 2x2 table analysis in R and show it is basically only 3 rows of syntax.

We show the same 2x2 table analysis in R and show it is basically only 3 rows of syntax. We discuss the default chi-square test in R which is the Yates correction which is similar to the Fisher's Exact test.

We continue to discuss Chi-squared analysis in R. I show that each test has a small instruction file that shows what each parameter in the syntax represents and what their defaults are. Some descriptions are better than others. There are also vignettes that show you examples of the syntax use.

This is a video on a short cut of the chi-squared test using aggregate data. You can set up a small table matrix and use Weight Cases in SPSS to do a Chi-squared analysis on aggregate data.

I had mentioned in another video that I gave the formula for the 2x2 table z test of two proportions and I mentioned that the chis squared could be used to analyze the same table. I told you it is because the z^2 distribution is equivalent to the chi-squared distribution. I show you here the value of the z-test for 2 proportions is in fact the square of the test statistic for the chi-squared.

This is a continuation of showing the z^2 distribution is equivalent to the chi-squared distribution. Also just for your information you can use chi-squared test to test two variances.

Here we go over the sample size calculation for the chi-squared 2x2 table test. I also offer a semi-formal proof showing how the three are statistically equivalent that my mentor Dr Robert Duncan gave me.

We begin the study of the stratified analysis for 3 categorical variables analysis, the Cochran-Mantel Haenszel test (CMH). The CMH is used to adjust out of the eefect of a third variablefrom the measure of the original association, i.e. the association between exposure and outcome. We first calculate the proportions is the table to help students get comfortable with the calculations. We do it for the 3 tables created by stratifying the crude table by a categorical variable with 3 categories. I'm doing a short version of the calculations so students can check their work as they watch the video. You should use all the decimal places carried through the analysis in the real analysis.

We calculate the odds ratios for the three tables, again, to make the students comfortable with doing these calculations. Remember I'm doing a short version of the calculations so students can check their work as they watch the video. You should use all the decimal places carried through the analysis in the real analysis.

We calculate even more the odds ratios for the three tables, again, to make the students comfortable with doing these calculations. Remember I'm doing a short version of the calculations so students can check their work as they watch the video. You should use all the decimal places carried through the analysis in the real analysis.

We define what the CMH tests for and we talk about some of the different scenarios of different outcomes. The non stratified analysis may have an effect of a third variable influencing (biasing) the association. The CMH gives you a stratum-adjusted measure of the original association, i.e. the association between exposure and outcome. We also go over the process models which need to be addressed when more than 2 variables are being considered in the inference. The process model are graphs that show how the different variables can be related to each other and how they lead to different interpretations of the inference. You can have a model that is along one chain of variables and in causality they call that the causal chain. We go over the different process models.

We show how the process model is indicated by different associations found to be significant in the statistical models. We then explain the CMH test statistic and what the variables in it are representative of in the data and describe the hypothesis test for the example.

Here we use R software, to thoroughly examine the influence of different configurations of the sample size calculation. We also influence the size of different values in the configurations to gain a better understand of the parameters used to calculate the sample size for the Cochran-Mantel-Haenszel test for stratified tables. Remember that you are calculating the sample size before the study starts. You should look at different configurations and projected proportions you may obtain in the actual study you are going to use.

We continue to explore here, using R software, the influence of different configurations and different size of different values in the configurations for the sample size for the Cochran-Mantel-Haenszel test for stratified tables. There are two ways of specifying the sizes in the tables you are considering for your study.

Showing the process that you can employ to determine different effect sizes and different sample sizes for an up-coming study. You are trying to decide with data that you gathered to be able to calculate the sample size. Also different ways of reporting the sample size investigations you have done for the up-coming study.

Showing the different ways of reporting in the protocol the sample size investigations you have done for the up-coming study. You can put in the protocol either one straight forward number per group, or you can put a table of different sample sizes for a range of odds ratios and proportions that might occur in the study.

We begin looking at a incredibly messy data file someone sent me once as an example to start the discussion about data cleaning. There has been a lot of discussion in the past that data cleaning can take up as much as 80% of the time of a working statisticians time.

We start looking at the file and all its problems: variable names, empty columns, text recoding error.

We discuss in data cleaning the issue of text fields with more than 10 or 15 categories and how to deal with it.

We talk about combining text fields for data cleaning. Talk about making separate variables for categorical text variables. These are considered dummy variables.

We discuss how skip patterns effect the data variables and how to use those in the analysis. Recoding variables.

You have to check the data for values out of range. Also, there may be errors in units. Sometime one person data entering in milligram per liter (mg/L) and the other is entering data in microgram per liter (ug/L). ALso if there is a group with missing in a variable you won't be able to compare groups. Also, different databases can put different codes in a data file. Also how to deal with medical notes.

We are starting to do analysis from start to finish and the different file types that are available in medical records. Also how to change from long to wide data analysis file types. We use SPSS.

We start the analysis for a paired t-test. We have an assumption with the paired t-test and we explore that first before doing the analysis. We perform the paired t-test in SPSS.

We use a different data file to perform the two independent sample t0test. I show a way to number multiple rows in excel. We carry out the t-test in SPSS and interpret the output.

We use SPSS crosstabs for a chi-squared analysis. I show a setting that will give the odds ratio and its confidence interval. We discuss the outcome and how to report it.

Analyzing a retrospective study. We use the same chi-squared but instead of reporting the rate of sick ffor unexposed versus the rate of sick in the unexposed, we report the rate of exposed in the sick and the rate of exposed in the not sick. I explained this earlier in the chi-squared video. We discuss the output and its interpretation.

We run a Cochran Mantel Haenszel analysis. We test is treatment associated with drug effect status and we were going to stratify by the participants health status. We looked at the outcome and interpret the results.

This is a simple crosstab test analysis using Julius AI. That was a Chi-Squared test of income with GetHIV and they gave you a p-value and I did it in SPSS and that is what the p-value is. For it to give you a good interpretation you have to give it more information about the variables: such as what is the variable GetHIV measuring. It can assume what income is. But it can give you a simple statistical interpretation stating that since the p-value is not <0.05 there is no association of income with GetHIV.