课堂笔记(3) 假设检验 Hypothesis testing

Basic knowledge

Estimated standard error of βˆi

Hypothesis test for the slope parameter βi

R语言例子

Ex1：海湾捕鱼 Gulf Menhaden catch data

Using the pairs( ) function to informally judge the appropriateness of fitting linear relationships.

R outputs relevant to testing βi

T test: slope parameter β1 for vessels

Special case：Testing H0 : β1 = a in simple regression

Lab

Basic knowledge

Using the ANOVA table we can test the hypothesis : H0 :β1 =β2 =...=βp =0, but this F-test is of limited use.

多元线性回归中，回归方程显著并不意味着每个自变量x对y影响都显著，所以要对每个自变量进行显著性检验。

When we consider hypothesis testing for specific slope parameters. In particular, how to test H0 : βj = 0. We will use the t-test for regression estimates.

Estimated standard error of βˆi

The values βˆ are estimates of the true (and unknown) β.

The denominator 分母 ${\color{Blue} \sqrt{Var\left ( \widehat{\beta i} \right )}}$ is usually denoted ${\color{Blue} e.s.e\left ( \widehat{\beta i} \right )}$ , and is called the

estimated standard error of ${\color{DarkRed} \widehat{\beta i}}$ and is easily obtained using R.

$\frac{\widehat{\beta i}-\beta i}{e.s.e\left ( \widehat{\beta i} \right )}\sim t_{n-p-1}$

Hypothesis test for the slope parameter βi

$H0:\beta i=a$ vs $H0:\beta i\neq a$ Where a is a constant value.

The associated test statistic is

${\color{Blue} t^{*}=\frac{\widehat{\beta i}-a}{e.s.e\left ( \widehat{\beta i} \right )}}$

As with all hypothesis tests, we never accept anything. We only reject/do no reject the null H0
The most commonly, we test H0 : a = 0.

Rejection region for two-sided (most common) test α = 0.05

Rejection region for one-sided test (much less common) α = 0.05

R语言例子
Ex1：海湾捕鱼 Gulf Menhaden catch data

# Gulf Menhaden fish data
year     <- 1964:1979
# 1964到1979的16年的数据
catch    <- c(409.4, 463.1, 359.1, 317.3, 373.5, 523.7, 548.1, 728.2,501.7, 486.1, 578.6, 542.6, 561.2, 447.1, 820.0, 777.9)
vessels  <- c(76, 82, 80, 76, 69, 72, 73, 82, 75, 65,71, 78, 81, 80, 80, 77)
pressure <- c(282.9, 335.6, 381.3, 404.7, 382.3, 411.0, 400.0, 472.9,447.5, 426.2, 485.5, 536.9, 575.9, 532.7, 574.3, 533.9)
fish.dat <- data.frame(year, catch, vessels, pressure)# 使用head函数获取dataframe的前几行数据
head(fish.dat)
##   year catch vessels pressure
## 1 1964 409.4  76    282.9
## 2 1965 463.1  82    335.6
## 3 1966 359.1  80    381.3
## 4 1967 317.3  76    404.7
## 5 1968 373.5  69    382.3
## 6 1969 523.7  72    411.0

First step in a multiple regression analysis should always be to plot the data. We should first satisfy data that a linear model is appropriate at all.（画图分析模型是否正确）
In multiple regression it makes sense to plot all pairs of scatter plots, ie. plotting each variable against every other variable.（画成对的散点图）

Using the pairs( ) function to informally judge the appropriateness of fitting linear relationships.

This has the added benefit of showing us whether certain independent variables are correlated with each other.

?pairs
## Scatterplot Matrices 散点图矩阵
## Description: A matrix of scatterplots is produced.产生了一个散点矩阵。pairs(fish.dat)
# matrix of scatterplots for informally checking linear associations
# 用于非正式检查线性关联的散点矩阵

在此图中：你可以看到所有指定变量间的二元关系。例如year和catch的散点图可在两变量的行列交叉处找到。值得注意的是，主对角线的上方和下方的六幅散点图是相同的。

From the plots:
The relationships look plausibly linear, or at least there is nothing to suggest otherwise.
Fit a linear regression model:
We can proceed to fit a linear model for the total catch, in terms of the independent variables vessels and pressure. We are estimating catch = β0 + β1vessels + β2pressure + ε

# fit a linear model for catch in terms of vessels and pressure
fish.lm <- lm(catch ~ vessels + pressure, data = fish.dat)
summary(fish.lm)

R outputs relevant to testing βi

T test: slope parameter β1 for vessels

$H0:\beta 1=0$ vs $H0:\beta 1\neq 0$

The test statistic for this null hypothesis is

$t^{*}=\frac{\widehat{\beta 1}-0}{e.s.e\left ( \widehat{\beta i} \right )}=\frac{2.2783}{6.387}=0.357$

Compare with critical value $t_{n-p-1,\alpha /2}=t_{16-2-1,0.05/2}=t_{13,0.025}=2.16$

To obtain $t_{13,0.025}$ in R, use the command qt( )

qt(p = 1 - 0.025, df = 13)
# p= 1 − α/2
## [1] 2.160369

Assume this is a two-sided test, and set α = 0.05
Since |t∗| = 0.357 < 2.16 = t13,0.025, do not reject H0.

We can also Testing H0 : βi = 0 using the p-value :

At the α level, Reject H0 if p<α. Do not reject H0 if p≥α.

Use of the p-value is completely equivalent to using a critical level and comparing it with the observed t∗ statistic.

① Summarise the derived regression equation

catch = −119.09 + 2.278 × vessels + 1.054 × pressure + ε

The positive estimates for β1,β2 indicate that catch increases with number of vessels and fishing intensity. By 2278 tons for every extra vessel, and 1054 tons for each extra unit of intensity.

② Obtain likely catch when vessels = 80 and pressure = 400 catch80

400 = −119.09 + (2.278 × 80) + (1.054 × 400) = 484.8

So the estimated annual catch would be 484,800 tons under this scenario.

Special case：Testing H0 : β1 = a in simple regression

In simple regression, e.s.e(βˆ1) is easier to calculate

${\color{Blue} e.s.e\left ( \widehat{\beta i} \right )=\sqrt{\frac{\widehat{\sigma }^{2}}{SS_{xx}}}}$

The test statistic t∗ therefore has the following simplified form

${\color{Blue} t^{*}=\frac{\widehat{\beta 1}-a}{e.s.e\left ( \widehat{\beta i} \right )}=\frac{\widehat{\beta 1}-a}{\sqrt{\widehat{\sigma }^{2}/SS_{xx}}}}$

参考书籍：

Montgomery, D.C., Peck, E.A., Vining, G.G. (2012). Introduction to Linear Regression Analysis (5th Edition). Wiley. (ILRA)

何晓群刘文卿. 应用回归分析第五版. 中国人民大学出版社.

John O. Rawlings, Sastry G. Pantula, David A. Dickey. Applied Regression Analysis: A Research Tool.

Robert I. Kabacoff. R语言实战第二版. (图灵程序设计丛书)