How can you explain the concept of expected frequencies in chi-square tests?

Expected frequencies in a chi-square test are the theoretical values you would expect to observe in each category if the null hypothesis were true. They are calculated based on the theoretical distribution of the data, considering the row and column totals for contingency tables or the expected distribution for each category in a goodness-of-fit test. The chi-square test compares these expected frequencies with the observed frequencies (the actual data) to determine if there is a statistically significant difference between them.

In the context of the Chi-square test for independence, expected values are the theoretical frequencies calculated by multiplying the marginal totals for each row and column and then dividing by the total sample size. These calculations assume the categories are independent, which forms the basis of the null hypothesis. The expected values proportionally reflect the overall distribution of each category in the sample.

In chi-square tests, expected frequencies play a crucial role. Here's a concise summary: 1. Chi-Square Goodness of Fit Test: - Compares observed frequencies with expected frequencies based on a hypothesized distribution. - Helps assess whether a categorical variable follows the expected pattern. 2. Chi-Square Test of Independence: - Examines association between two categorical variables. - Compares observed and expected frequencies in a contingency table.

Imagine a fair coin toss (null hypothesis: no bias). Expected heads (and tails) equals total tosses divided by two. Chi-square tests compare these expected frequencies (based on the null hypothesis) with actual counts (heads vs tails) to see if the real world deviates significantly, suggesting the null hypothesis might be wrong.

The expected values in a chi-square test refer to the values that would be expected if there were no significant difference between the observed and expected frequencies. Some key points about the expected values: - They are calculated based on the total sample size and the hypothesized distribution or proportions. - They represent the values that would be expected if the null hypothesis (no significant difference) were true. - The test compares the observed frequencies to these expected frequencies. - A large difference between observed and expected values leads to a larger chi-square statistic and a smaller p-value, indicating a significant difference. - The expected values should be greater than 5 for the chi-square test to be valid.

Calculating frequency is a fundamental step in statistical analysis and data interpretation. Frequency refers to the number of times a particular value or category appears in a dataset. It helps in summarizing and understanding the distribution of data, making it easier to identify patterns, trends, and outliers. There are different types of frequencies, including absolute frequency, relative frequency, cumulative frequency, and cumulative relative frequency, each serving a unique purpose in data analysis.

Here’s a step-by-step explanation of how to calculate expected frequencies: 1. Construct the Contingency Table. 2. Calculate Row and Column Totals. 3. Compute the Expected Frequencies. 4. Construct the Table of Expected Frequencies. 5. Compare Observed and Expected Frequencies. In the chi-square test, you compare the observed frequencies with the expected frequencies to determine if there is a significant difference between them. By comparing the observed data with the expected frequencies, you can determine whether the observed distribution significantly deviates from what would be expected under the assumption of independence, helping to identify whether there is an association between the variables.

This is a probabilistic output of observations that follow the said theoretical distribution. The total frequency of sample is multiplied with probablity of each event to happen giving out expected frequency for that event. In case of, contingency table no. of observations in rows* no. of observations in columns divided by total observations giving expected count for each cell in table.

This explanation is clear, and adding an example can make it more relatable. Imagine you have a survey of 100 people, with 60 women and 40 men, and you want to see if gender influences ice cream flavor preference (chocolate or vanilla). If 50 people like chocolate and 50 like vanilla, you'd calculate expected frequencies to see if the observed preferences match what you'd expect by chance. For women liking chocolate: (60 x 50)/100 = 30. For men liking chocolate: (40 x 50)/100 = 20. The same would be done with vanilla. This process helps highlight whether any observed patterns are due to actual preferences or random distribution. Understanding this step is crucial for interpreting chi-square test results effectively.

Chi- Square test statistic is also known as goodness of fit, that is to test how well data fits on a particular distribution. This is measure of deviation between observed and expected. Calculated by taking a square of deviation between observed and expected frequency w.r.t. expected frequency, then summing for each category. This gives out test statistic to be compared with chi-square probabilistic distribution table for w.r.t degrees of freedom and critical value.

Very accurate, and an example might help as well. Imagine you're running a survey to see if there's a link between favorite ice cream flavors and gender. After collecting your data, you calculate the expected frequencies, which are what you'd anticipate if there's no real connection between the two. Let's say you expect 30 women and 20 men to like chocolate based on the overall preference rate. Now, you compare these expected numbers to what you actually observed. If you find more women liking chocolate than expected, you'd use the chi-square statistic to see if this difference is just due to random chance or if it suggests a real pattern. The chi-square statistic helps us decide if our observed data is really surprising or what's expected.

The Chi-Squared test statistic only works on categorical variables. The way this is described as, "How much your observed frequencies deviate from the expected frequencies," is true but not for any random variable. The random variable must be of categorical or discrete in nature. The context of how much the observation deviates from the expectation is wholly based on the concept your assumption for what the distribution is. The definition of the Chi-Squared distribution is the sum of squared normal distributions and that makes it only relevant to normally distributed random variables that are categorical. Its critical that you find the correct test-statistic based on the dataset you have and not hastily assume normality.

Expected frequencies are the theoretical frequencies you would expect to observe if there is no association between variables, based on the null hypothesis. They are calculated by multiplying the row total by the column total and then dividing by the overall total for each cell in a contingency table. Degrees of freedom (df) in a chi-square test measure the number of values that can vary while maintaining the overall totals. For a contingency table, df is calculated as (number of rows - 1) multiplied by (number of columns - 1). This helps determine the appropriate chi-square distribution to compare against the test statistic.

This is more or less related to chi-square tests, but I wanted to chime in when talking about degrees of freedom. I don't think they are ever properly explained, you just calculate with them and don’t really question it. But I came across an example years ago explaining it with credit card fraud. We do know that it is basically the number of values in a calculation that are free to vary without breaking other constraints. But imagine a thief trying to guess a 3-digit CVV code on a credit card. If they know the first digit, they have fewer degrees of freedom left. And if they know the first two digits, they have even fewer degrees of freedom, making it much easier to guess the last digit. I find it very easy to explain like this!

In chi-square tests, degrees of freedom help determine the shape of the chi-square distribution, which is essential for interpreting the test results. Degrees of freedom impact the chi-square test’s critical value, which you use to determine if your observed results are due to chance or if there’s a significant effect. Higher degrees of freedom typically mean a wider distribution and can affect whether your test results are deemed statistically significant.

I guess this is the toughest of all to explain, the degree of freedom are no. of independent piece of information to calculate a statistic. In the case, say u have a 4*5 contingency table. Data for each cell could be calculated by (no. of obs rows* col)/total col. Now, after the 3rd row or 4th col. Independence of information in last row/col. Is limit due to the grand total. Hence, the degree of freedom is multiplication of no. of rows -1 * no. of col. -1 in above case become (4-1)*(5-1)=12

It refers to the number of values that can vary freely in the calculation of the test statistic. Specifically: - For a one-way chi-square test (testing a single categorical variable), the degrees of freedom is equal to the number of categories minus 1. - For a two-way chi-square test (testing the relationship between two categorical variables), the degrees of freedom is calculated as (number of rows - 1) * (number of columns - 1). The degree of freedom(dof) determines the critical value used to assess the statistical significance of the chi-square test statistic. A larger (dof) means the chi-square distribution has a higher spread, resulting in a larger critical value needed to declare statistical significance at a given alpha level.

Chi-Square is square of standard normal distribution, it's limit of integration goes from 0 to infinity. Hence, for different degrees of freedom and critical value a chi-square value is calculated and added in chi-square table. This is used to compare with chi-square test statistic value higher value suggest rejection of null hypothesis saying there is statistically significant difference between observed and expected frequencies while lower value helps in accepting null hypothesis.

Explanation is clear, and could benefit from emphasizing the practical significance of the findings. For instance, while statistical significance indicates that the observed differences are unlikely to be due to chance, it doesn't necessarily tell us how strong or meaningful those differences are in real-world terms. Adding context or examples can help readers understand the implications of their results better. Example: an ice cream survey, a statistically significant difference in flavor preferences between groups could suggest a marketing opportunity or need for more inclusive product offerings. Understanding both the statistical and practical significance of chi-square test results is key to making informed decisions based on the data.

I had a manager who would run SPSS tables for all variables by all variables, then using significant chi-square stats go into deeper dives. With computing power as it is today, that might not be a bad idea. Good research practices though tell us that we should use the chi-square to accept or reject the null hypothesis. If the statistic is significant, you can reject the null hypothesis (usually no difference) and accept the hypothesis being tested (males are more likely to have blue eyes).

Interpretar os resultados de um teste de qui-quadrado envolve comparar a estatística de qui-quadrado calculada com um valor crítico da distribui??o qui-quadrado, que depende dos graus de liberdade do teste (geralmente o número de categorias menos um para cada variável envolvida). Se a estatística de qui-quadrado for maior que o valor crítico, rejeita-se a hipótese nula de que as variáveis s?o independentes, indicando que há uma associa??o significativa entre as categorias analisadas. é importante também considerar o valor-p, que oferece uma medida da probabilidade de observar uma estatística de qui-quadrado t?o extrema quanto a calculada, assumindo que a hipótese nula é verdadeira.

To calculate the expected frequency for a specific combination of categorical variables multiply the column total by the row total , and divide by the sample size.

Understanding expected frequencies in chi-square tests is crucial across fields like biology, marketing, and psychology. They help gauge the significance of findings. Whether assessing a drug's efficacy or analyzing consumer trends, mastering these values enables informed decision-making from categorical data.

I like using the chi-square as the first test of significance because it is easy to calculate (yes, been there, done that) and can quickly separate significant from insignificant findings. SaS, SPSS, etc. will display the chi-square and the significance of it. The significance you want depends on what is being measured. For consumer research, smaller significance levels are acceptable, (.05, .001) for medical, and healthcare high levels (.0000) or more are desired. From here other statistics can be used to further draw conclusions from the data using advanced test like regression, ANOVA, etc.

Let's analyze a drug effectiveness by comparing it to another drug. Say, drug1 is 40% effective, and drug2 is 50%. Is drug2 really better? For an independence (chi-square) test we take these input data: Effective(Y) Not Effective(N) D1: 4 6 D2: 7 7 Under independence, P(D1 & Y) = P(D1)*P(Y)= (6+4)/24 * (7+7)/24. We compare that to the observed P(D1 & Y)= 4/24 by calculating the difference and then square it. We do this for P(D1 & N), P(D2 & N), and P(D2 & Y). We sum up the four squared differences - this is our chi-squared statistic. If observed P(D1 & Y) is equal to P(D1)*P(Y), and same is true for the other 3 terms, then the chi-square statistic = 0, and the drugs are "the same".

Expected frequencies in chi-square tests are the frequencies you would expect to find in each category of a contingency table if the null hypothesis were true. The null hypothesis typically states that there is no association between the variables. If the chi-square statistic is greater than the critical value, you reject the null hypothesis, indicating that there is an association between the variables. Expected frequencies are a crucial component in determining whether the observed data significantly deviates from what we would expect under the null hypothesis