Chi-square test of parameter equality là gì
You are interested in testing the equality of the proportion parameter of k number of populations with respect to binary random variable (e.g., Yes, No): Show
H0: P1 \= P2 \=.....= Pk Arrange the resulting frequencies in a 2 by k (or k by 2) crosstable, i.e., sample size, and number of Yes�s (or number of No�s) for each populations. Enter the frequencies into the following table by replacing "X" with your frequencies, starting from the upper left corner without leaving any gaps, and then click the Calculate. Non-entry rows and columns are not included in the calculations. In entering your data to move from cell to cell in the data-matrix use the Tab key not arrow or enter keys. For a person without a background in stats, it can be difficult to understand the difference between fundamental statistical tests (not to mention when to use them). Here are the differences between the most common tests, how to use null value hypotheses in these tests and the conditions under which you should use each particular test.
Before we learn about the tests, let’s dive into some key terms. Defining Our TermsNull Hypothesis and Hypothesis TestingBefore we venture into the differences between common statistical tests, we need to formulate a clear understanding of a null hypothesis. The null hypothesis proposes that no significant difference exists between a set of given observations. In other words:
To reject a null hypothesis, one needs to calculate test statistics, then compare the result with the critical value. If the test statistic is greater than the critical value, we can reject the null hypothesis. More From Built In Experts4 Probability Distributions Every Data Scientist Needs to Know Critical ValueA critical value is a point (or points) on the scale of the test statistic beyond which we reject the null hypothesis. We derive the level of significance ( Critical value can tell us the probability of two sample means belonging to the same distribution. The higher the critical value means the lower the probability of two samples belonging to the same distribution. The general critical value for a two-tailed test is 1.96, which is based on the fact that 95 percent of the area of a normal distribution is within 1.96 standard deviations of the mean. Critical values can be used to do hypothesis testing in the following ways:
If the test statistic is lower than the critical value, accept the null hypothesis; otherwise reject it. Learn more about calculating a critical value: Critical value (z*) for a given confidence level Note: Some statisticians would use instead of critical value for conducting null hypothesis. Sample vs. PopulationIn statistics, population refers to the total set of observations we can make. For example, if we want to calculate the average human height, the population will be the total number of people actually present on Earth. A sample, on the other hand, is a set of data collected or selected from a predefined procedure. For our example above, a sample is a small group of people selected randomly from different regions of the globe. To draw inferences from a sample and validate a hypothesis, the sample must be random. For instance, if we select people randomly from all regions on Earth, we can assume our sample mean is close to the population mean, whereas if we make a selection just from the United States, then our average height estimate/sample mean cannot be considered close to the population mean. Instead, it will only represent the data of a particular region (the United States). That means our sample is biased and is not representative of the population. More on Biased DataWhat's Wrong With Your Statistical Model? Skewed Data. DistributionAnother important statistical concept to understand is distribution. When the population is infinitely large, it’s not feasible to validate any hypothesis by calculating the mean value or test parameters on the entire population. In such cases, we assume a population is some type of a distribution. While there are many forms of distribution, the most common are binomial, Poisson and discrete. You must determine the distribution type to calculate the critical value and decide on the best test to validate any hypothesis. Now that we’re clear on population, sample and distribution, let’s learn about different kinds of tests and the distribution types for which they are used. Poisson Piqued Your Interest?The Poisson Process and Poisson Distribution, Explained (With Meteors!) Statistical TestsP-value, Critical Value and Test StatisticAs we know, critical value is the point beyond which we reject the null hypothesis. P-value, on the other hand, is the probability to the right of the respective statistic (z, t or chi). The benefit of using p-value is that it calculates a probability estimate, which means we can test at any desired level of significance by comparing this probability directly with the significance level. For example, assume the z-value for a particular experiment comes out to be 1.67 which is greater than the critical value at five percent (1.64). Now, to check for a different significance level of one percent, we calculate a new critical value. However, if we calculate p-value for 1.67 and it comes to be 0.047, we can use this p-value to reject the hypothesis at a five percent significance level since 0.047 < 0.05. However, with a more stringent significance level of one percent, we’ll fail to reject the hypothesis since 0.047 > 0.01. It’s important to note here that there’s no double calculation required. Learn about the t-test, the chi square test the p-value and more Z-TestIn a z-test, we assume the sample is normally distributed. A z-score is calculated with population parameters such as population mean and population standard deviation. We use this test to validate a hypothesis that states the sample belongs to the same population.
The statistic used for this hypothesis testing is called z-statistic, the score for which we calculate as:
If the test statistic is lower than the critical value, accept the hypothesis. T-TestWe use a t-test to compare the mean of two given samples. Like a z-test, a t-test also assumes a normal distribution of the sample. When we don’t know the population parameters (mean and standard deviation), we use t-test. The Three Versions of a T-Test
The statistic for this hypothesis testing is called t-statistic, the score for which we calculate as:
There are multiple variations of the t-test. Note: This article focuses on normally distributed data. You can use z-tests and t-tests for data which is non-normally distributed as well if the sample size is greater than 20, however there are other preferable methods to use in such a situation. Chi-Square TestWe use the chi-square test to compare categorical variables. The Two Types of Chi-Square Test
A small chi-square value means that data fits. A large chi-square value means that data doesn’t fit. The hypothesis we’re testing is:
The statistic used to measure significance, in this case, is called chi-square statistic. The formula we use to calculate the statistic is: `z = (x — μ) / (σ / √n)`0 where `z = (x — μ) / (σ / √n)`1=observed frequency count at level r of Variable A and level c of Variable B `z = (x — μ) / (σ / √n)`2=expected frequency count at level r of Variable A and level c of Variable B T-Test vs. Chi-SquareWe use a t-test to compare the mean of two given samples but we use the chi-square test to compare categorical variables. Built In Tutorials for Data ScientistsA Primer on Model Fitting ANOVAWe use analysis of variance (ANOVA) to compare three or more samples with a single test. The Two Major Types of ANOVA
The hypothesis we’re testing with ANOVA is:
The statistics used to measure the significance in this case are F-statistics. We calculate the F-value using the formula: `z = (x — μ) / (σ / √n)`3, where `z = (x — μ) / (σ / √n)`4=residual sum of squares `z = (x — μ) / (σ / √n)`5=number of restrictions `z = (x — μ) / (σ / √n)`6=number of independent variables There are multiple tools available such as SPSS, R packages, Excel etc. to carry out ANOVA on a given sample. The TakeawayIf you learn only one thing from this article, let it be this: In all of these tests we’re comparing a statistic with a critical value to accept or reject a hypothesis. However, the statistic and the way to calculate it differ depending on the type of variable, the number of samples you’re analyzing and whether or not we know the population parameters. We can thus choose a suitable statistical test and null hypothesis. This principle is instrumental to understanding these basic statistical concepts. |