Without other qualification, 'chi-squared test' often is used as short for Pearson's chi-squared test.A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.
Other chi-squared tests
- Cochran–Mantel–Haenszel chi-squared test.
- McNemar's test, used in certain 2 × 2 tables with pairing
- Tukey's test of additivity
- The portmanteau test in time-series analysis, testing for the presence of autocorrelation
- Likelihood-ratio tests in general statistical modelling, for testing whether there is evidence of the need to move from a simple model to a more complicated one (where the simple model is nested within the complicated one).
wald test
Test on a single parameter
In the univariate case, the Wald statistic is
which is compared against a chi-squared distribution.
Alternatively, the difference can be compared to a normal distribution. In this case the test statistic is
where 
is the standard error of the maximum likelihood estimate (MLE). A reasonable estimate of the standard error for the MLE can be given by 
, where
is the Fisher information of the parameter.
Test(s) on multiple parameters
The Wald test can be used to test a single hypothesis on multiple parameters, as well as to test jointly multiple hypotheses on single/multiple parameters. Let 
be our sample estimator of P parameters (i.e., is a Px1 vector), which is supposed to follow asymptotically a normal distribution with covariance matrix V, 
. The test of Q hypotheses on the P parameters is expressed with a Q x P matrix R:
The test statistic is:
![(R\hat{\theta}_n-r)^{'}[R(\hat{V}_n/n)R^{'}]^{-1}(R\hat{\theta}_n-r) \quad \xrightarrow{\mathcal{D}}\quad \Chi^2_Q](/image/aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jODliNjdlNWM3ZDc1YzhmNjViMDVkYzk0YzFjOTkzNWQwZGYwYzg1.png)
where 
is an estimator of the covariance matrix.
t test 和方差分析要檢驗的平均數的差異，變量為連續變量，變量為漸近正態分布，而卡方檢驗不要求分布。
卡方檢驗 變量為類別變量，檢驗的是不同類別的數據是否在另一個變量上一致，如果不一致則這兩個變量具有相關性，一致說明沒有相關性，類似於交互作用的意思。
當進入回歸分析的自變量為啞變量時，該自變量在該回歸中，就類似於t檢驗，其系數值為平均差值，系數的檢驗采用的是t檢驗。
自由度為n-1的t分布 的平方等於自由度（1，n-1）F分布。
自由度為m-1的卡方/n-m-1的卡方分布為（m-1，n-m-1）F分布。
實際上t分布就是 自由度 1的卡方/自由度為n-1的卡方分布。
t檢驗的平方是( x平均-總體平均u)^2/標准誤^2。。
標准誤^2服從自由度n-1卡方分布。
（x平均-總體平均u）服從自由度（2-1）=1的卡方分布， (n-1)自由度t^2=F自由度（1，n-1）。
n足夠大 t分布近似正態分布
2組樣本下n不夠大t分布為自由度（1，n-1）F分布。
卡方分布就是標准誤^2分布。
多樣本下分布自由度（m-1，n-1）F分布就是方差分析。
還可以得出一元線性回歸的t檢驗 的平方為F檢驗，並與F的方差分析等價。
多元線性回歸就是多因素方差分析等價。
n足夠大是z或者u檢驗，或，t檢驗自由度n-1足夠大t=u是一樣的為正態分布，n不夠大就服從t檢驗，卡方檢驗是對標准誤的平方檢驗，信息量小於t檢驗，所以精確性小於t檢驗，這就是為什么計數資料結果是率0-1之間並且方差大，用t檢驗或u檢驗需要樣本大，所以用卡方檢驗只看方差時就可以檢驗，但是卡方檢驗的精確性差了，加強精確性可以用logistic回歸。
總之u檢驗，t檢驗，F檢驗，卡方檢驗，一元線性回歸，多元性回歸在一定條件下互相轉化！
及對於大樣本u檢驗，就是有多個自變量的多元線性回歸就是多因素協方差分析，只有一個自變量多元線性回歸變為一元線性回歸，自變量x有3個或以上的值就是多樣本單因素的方差分析，只有2個取值，就是2個樣本單因素方差分析，就是F（1，n-1）檢驗，這個分布開平方就是t（n-1）檢驗，n足夠大所以就是u檢驗！這就是基礎統計檢驗的關系。
F-test
Exact "F-tests" mainly arise when the models have been fitted to the data using least squares.The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.
Common examples of F-tests
Common examples of the use of F-tests include the study of the following cases:
- The hypothesis that the means of a given set of normally distributed populations, all having the same standard deviation, are equal. This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA).
- The hypothesis that a proposed regression model fits the data well. See Lack-of-fit sum of squares.
- The hypothesis that a data set in a regression analysis follows the simpler of two proposed linear models that are nested within each other.
In addition, some statistical procedures, such as Scheffé's method for multiple comparisons adjustment in linear models, also use F-tests.
F-test of the equality of two variances
The F-test is sensitive to non-normality. In the analysis of variance (ANOVA), alternative tests include Levene's test, Bartlett's test, and the Brown–Forsythe test. However, when any of these tests are conducted to test the underlying assumption of homoscedasticity (i.e. homogeneity of variance), as a preliminary step to testing for mean effects, there is an increase in the experiment-wise Type I error rate.