1. Origin of T-test and F-test
In general, in order to determine the probability of error in inferring from a sample statistical result to the population, we use some statistical methods developed by statisticians to perform statistical tests.
By comparing the obtained statistical test value with the probability distribution of some random variable established by the statistician, we can know what % of the chances we will get the current result. If, after comparison, it is found that the probability of occurrence of this result is very small, that is to say, it occurs under very rare and rare occasions; then we can confidently say that this is not a coincidence, but a statistical It is meaningful (in statistical terms, it is able to reject the null hypothesis, Ho). On the contrary, if the comparison shows that the probability of occurrence is high and not uncommon; then we cannot confidently point out that it is not a coincidence, maybe it is a coincidence, maybe not, but we are not sure.
F-values and t-values are these statistical test values, and their corresponding probability distributions are F-distribution and t-distribution. Statistical significance (sig) is the probability of an outcome in the current sample.
2. Statistical significance (P value or sig value)
The statistical significance of the results is an estimate of how true the results are (representative of the population). Professionally, the p-value is a decreasing indicator of the credibility of the results. The larger the p-value, the less we can think that the correlation of variables in the sample is a reliable indicator of the correlation of variables in the population. The p-value is the probability of error that an observation is considered valid, that is, representative of the population. For example, p=0.05 suggests that there is a 5% chance that the association of variables in the sample is due to chance. That is, assuming that there is no correlation between any variables in the population, if we repeat similar experiments, we will find that in about 20 experiments, the correlation of the variables we study will be equal to or stronger than our experimental results. (This is not to say that if there is an association between variables, we can get the same result 5% or 95% of the time. When variables in a population are associated, the likelihood of repeating the study and finding an association is related to the statistical power of the design.) In many fields of research, a p-value of 0.05 is often considered the borderline level of acceptable error.
3. T-test and F-test
As for the specific content to be checked, it depends on which statistical procedure you are doing.
As an example, for example, you want to test whether the difference between the means of two independent samples can be inferred to the population, and the t-test is performed.
The mean of a variable (such as height) in the two samples (such as boys and girls in a certain class) is not the same, but can this difference be inferred to the overall population, and is there a difference in the overall situation? Could it be that there is no difference between male and female students in the population, but that the values of these 2 samples are different because you are so lucky?
To do this, we perform a t-test and calculate a t-test value.
Compare with the t-distribution of random variables based on "no difference in the population" established by statisticians to see what % chance (ie, the significance sig value) of the current result will be obtained.
If the significance sig value is small, say <0.05 (less than 5% chance), that is, "if" there is "true" no difference in the population, then only if the chance is very small (5%), very rare Under these circumstances, the current situation will occur. Although there is still a 5% chance of error (1-0.05=5%), we can still say with "more confidence" that this situation in the current sample (the difference between male and female students) is not a coincidence, it is statistically significant , the null hypothesis that "there is no difference between boys and girls in the population" should be rejected. In short, there should be differences in the population.
The test content of each statistical method is different. It is also a t-test, which may be whether there is a difference in the above-mentioned test population, or whether a single value in the test population is equal to 0 or equal to a certain value.
As for the F-test, analysis of variance (or analysis of variance, Analysis of Variance), its principle is roughly the same as above, but it is carried out by examining the variance of variables. It is mainly used for: significance test of mean difference, separation of relevant factors and estimation of their effect on total variation, analysis of interaction between factors, equality of variance (Equality of Variances) test, etc.
4. Relationship between T-test and F-test
The t-test procedure is to test the significance of the difference between the means of the two samples. However, the t-test needs to know whether the variances of the two populations (Variances) are equal; the calculation of the t-test value will vary depending on whether the variances are equal. That is, the t-test depends on the Equality of Variances results. Therefore, while SPSS is conducting t-test for Equality of Means, it also needs to do Levene's Test for Equality of Variances.
1. In the column of Levene's Test for Equality of Variances, the F value is 2.36, and the Sig. is .128, indicating that there is "no significant difference" in the homogeneity test, that is, the two variances are equal (Equal Variances), so the following t test In the results table, you need to see the data in the first row, that is, the results of the t-test under the condition of equal variance.
2. In t-test for Equality of Means, the first row (Variances=Equal): t=8.892, df=84, 2-Tail Sig=.000, Mean Difference=22.99 Since Sig=.000, also That is, the difference between the means of the two samples is significant!
3. Which sig in the column of Levene's Test for Equality of Variances should you look at, or the Sig. (2-tailed) in t-test for Equality of Means?
The answer is: look at both.
Let’s first look at Levene’s Test for Equality of Variances. If the homogeneity of variance test is “no significant difference”, that is, the two variances are equal (Equal Variances), so the result table of the next t test depends on the data in the first row, that is Results of the t-test with equal variances.
Conversely, if the test for the homogeneity of variances is "significantly different", that is, the two variances are unequal (Unequal Variances), so the next t-test result table should look at the data in the second row, that is, the t in the case of unequal variances. the result of the inspection.
4. You are doing a T test, why is there an F value?
It is because to evaluate whether the variances (Variances) of the two populations are equal, we need to do Levene's Test for Equality of Variances, and we need to test the variance, so there is an F value.
Another explanation:
There are one-sample t-test, paired t-test and two-sample t-test.
One-sample t-test: It compares the unknown population mean represented by the sample mean with the known population mean to observe the difference between this group of samples and the population.
Paired t-test: It is a paired design method to observe the following situations: 1, two homogeneous subjects receive two different treatments; 2, the same subject receives two different treatments; 3, the same subject Before and after object processing.
The F test is also called the homogeneity of variance test. The F-test is used in the two-sample t-test.
Samples are randomly selected from the two research populations. When comparing the two samples, it is first necessary to determine whether the variances of the two populations are the same, that is, the homogeneity of variances. If the variances of the two populations are equal, the t test can be used directly. If they are not equal, the t" test or variable transformation or rank sum test can be used.
The F test can be used to judge whether the two population variances are equal.
If it is a single-group design, a standard value or an overall mean must be given, and at the same time, a set of quantitative observation results must be provided. The precondition for applying the t-test is that the group of data must obey a normal distribution; if it is a paired design, the difference between each pair of data. Values must obey a normal distribution;
If it is a group design, the individuals are independent of each other, and the two groups of data are taken from a normally distributed population and satisfy the homogeneity of variance. The reason why these preconditions are needed is that the t statistic calculated under such premise must obey the t distribution, and the t test is a test method based on the t distribution as its theoretical basis.
Simply put, the practical T test is conditional, one of which is to meet the homogeneity of variance, which needs to be verified by the F test
Statistical significance (p value)
The statistical significance of an outcome is an estimate of how true the outcome is (representative of the population). Professionally, the p value is a decreasing index of the credibility of the results. The larger the p value, the less we can think that the correlation of variables in the sample is a reliable indicator of the correlation of variables in the population. The p-value is the probability of error that an observation is considered valid, that is, representative of the population. For example, p=0.05 indicates that there is a 5% chance that the variable association in the sample is due to chance. That is, assuming that there is no correlation between any variables in the population, if we repeat similar experiments, we will find that in about 20 experiments, the correlation of the variables we study will be equal to or stronger than our experimental results. (This is not to say that if there is an association between variables, we can get the same result 5% or 95% of the time. When variables in a population are associated, the likelihood of repeating the study and finding an association is related to the statistical power of the design.) In many fields of research, a p-value of 0.05 is often considered the borderline level of acceptable error.
How to determine if the results are truly significant
Determining what level of significance is statistically significant in the final conclusion is inevitably arbitrary. In other words, the choice of the level at which the result is considered invalid and rejected is arbitrary. In practice, the final decision often depends on whether the data set comparison and analysis process results in a priori or just a pairwise comparison of means, on the amount of supporting evidence in the overall data set for consistent conclusions, and on from previous practice in this field of study. Typically, in many fields of science a result that yields a p-value ≤ 0.05 is considered the borderline of statistical significance, but this level of significance also includes a fairly high probability of error. Results 0.05≥p>0.01 were considered statistically significant, while 0.01≥p≥0.001 was considered highly statistically significant. Note, however, that this categorization is only an informal judgmental routine based on research.
Are all test statistics normally distributed?
Not exactly, but most tests are directly or indirectly related to it and can be derived from a normal distribution, such as a t-test, f-test, or chi-square test. These tests generally require that the analyzed variables are normally distributed in the population, that is, to satisfy the so-called normality assumption. Many observed variables are indeed normally distributed, which is why the normal distribution is a fundamental feature of the real world. Problems arise when one analyzes data on non-normally distributed variables with tests built on the normal distribution (see Normality Tests for Nonparametric and ANOVA).
There are two methods under this condition: one is to use an alternative nonparametric test (ie, no distribution test), but this method is inconvenient, because from the form of the conclusions it provides, this method is statistically inefficient and inefficient. flexible. Another approach is to use a test based on a normal distribution when it is determined that the sample size is large enough. The latter method is based on a rather important principle, which is extremely important for population tests based on the normal equation. That is, as the sample size increases, the shape of the sample distribution tends to be normal, even if the distribution of the variable under study is not normal.