### 统计代写|Generalized linear model代考广义线性模型代写|Null Hypothesis Statistical Significance Testing

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|Generalized linear model代考广义线性模型代写|Null Hypothesis Statistical Significance Testing

The main purpose of this chapter is to transition from the theory of inferential statistics to the application of inferential statistics. The fundamental process of inferential statistics is called null hypothesis statistical significance testing (NHST). All procedures in the rest of this textbook are a form of NHST, so it is best to think of NHSTs as statistical procedures used to draw conclusions about a population based on sample data.
There are eight steps to NHST procedures:

1. Form groups in the data.
2. Define the null hypothesis $\left(\mathrm{H}_{0}\right)$. The null hypothesis is always that there is no difference between groups or that there is no relationship between independent and dependent variables.
3. Set alpha $(\alpha)$. The default alpha $=.05$.
4. Choose a one-tailed or a two-tailed test. This determines the alternative hypothesis $\left(\mathrm{H}_{\mathrm{l}}\right)$.
5. Find the critical value, which is used to define the rejection region. Calculate the observed value.
6. Compare the observed value and the critical value. If the observed value is more extreme than the critical value, then the null hypothesis should be rejected. Otherwise, it should be retained.
7. Calculate an effect size.
8. Readers who have had no previous exposure to statistics will find these steps confusing and abstract right now. But the rest of this chapter will define the terminology and show how to put these steps into practice. To reduce confusion, this book starts with the simplest possible NHST: the $z$-test.

## 统计代写|Generalized linear model代考广义线性模型代写|z-Test

Recall from the previous chapter that social scientists are often selecting samples from a population that they wish to study. However, it is usually impossible to know how representative a single sample is of the population. One possible solution is to follow the process shown at the end of Chapter 6 where a researcher selects many samples from the same population in order to build a probability distribution. Although this method works, it is not used in the real world because it is too expensive and time-consuming. (Moreover, nobody wants to spend their life gathering data from an infinite number of populations in order to build a sampling distribution.) The alternative is to conduct a $z$-test. A z-test is an NHST that scientists use to determine whether their sample is typical or representative of the population it was drawn from.

In Chapter 6 we learned that a sample mean often is not precisely equal to the mean of its parent population. This is due to sampling error, which is also apparent in the variation in mean values from sample to sample. If several samples are taken from the parent population, the means from each sample could be used to create a sampling distribution of means. Because of the principles of the central limit theorem (CLT), statisticians know that with an infinite number of sample means, the distribution will be normally distributed (if the $n$ of each sample is $\geq 25$ ) and the mean of means will always be equal to the population mean.

Additionally, remember that in Chapter 5 we saw that the normal distribution theoretically continues on from $-\infty$ to $+\infty$. This means that any $\bar{X}$ value is possible. However, because the sampling distribution of means is tallest at the population mean and shortest at the tails, the sample means close to the population mean are far more likely to occur than sample means that are very far from $\mu$.

Therefore, the question of inferential statistics is not whether a sample mean is possible but whether it is likely that the sample mean came from the population of interest. That requires a researcher to decide the point at which a sample mean is so different from the population mean that obtaining that sample mean would be highly unlikely (and, therefore, a more plausible explanation is that the sample really does differ from the population). If a sample mean $(\bar{X})$ is very similar to a population mean ( $\mu$ ), then the null hypothesis $(\bar{X}=\mu)$ is a good model for the data. Conversely, if the sample mean and population mean are very different, then the null hypothesis does not fit the data well, and it is reasonable to believe that the two means are different.

This can be seen in Figure 7.1, which shows a standard normal distribution of sampling means. As expected, the population mean $(\mu)$ is in the middle of the distribution, which is also the peak of the sampling distribution. The shaded regions in the tails are called the rejection region. If, when we graph the sample mean, it falls within the rejection region, the sample is so different from the mean that it is unlikely that sampling error alone could account for the differences between $\bar{X}$

and $\mu$ – and it is more likely that there is an actual difference between $\bar{X}$ and $\mu$. If the sample mean is outside the rejection region, then $\bar{X}$ and $\mu$ are similar enough that it can be concluded that $\bar{X}$ is typical of the population (and sampling error alone could possibly account for all of the differences between $\bar{X}$ and $\mu$ ).

Judging whether differences between $\bar{X}$ and $\mu$ are “close enough” or not due to just sampling error requires following the eight steps of NHST. To show how this happens in real life, we will use an example from a UK study by Vinten et al. (2009).

## 统计代写|Generalized linear model代考广义线性模型代写|Cautions for Using NHSTs

NHST procedures dominate quantitative research in the behavioral sciences (Cumming et al., 2007; Fidler et al., 2005; Warne, Lazo, Ramos, \& Ritter, 2012). But it is not a flawless procedure, and NHST is open to abuse. In this section, we will explore three of the main problems with NHST: (1) the possibility of errors, (2) the subjective decisions involved in conducting an NHST, and (3) NHST’s sensitivity to sample size.

Type I and Type II Errors. In the Vinten et al. (2009) example, we rejected the null hypothesis because the $z$-observed value was inside the rejection region, as is apparent in Figure $7.4$ (where the z-observed value was so far below zero that it could not be shown on the figure). But this does not mean that the null hypothesis is definitely wrong. Remember that theoretically – the probability distribution extends from $-\infty$ to to . Therefore, it is possible that a random sample could have an $\bar{X}$ value as low as what was observed in Vinten et al.’s (2009) study. This is clearly an unlikely event – but, theoretically, it is possible. So even though we rejected the null hypothesis and the $\bar{X}$ value in this example had a very extreme z-observed value, it is still possible that Vinten et al. just had a weird sample (which would produce a large amount of sampling error). Thus, the results of this z-test do not prove that the anti-seizure medication is harmful to children in the womb.

Scientists never know for sure whether their null hypothesis is true or not – even if that null is strongly rejected, as in this chapter’s example (Open Science Collaboration, 2015; Tukey, 1991). There is always the possibility (no matter how small) that the results are just a product of sampling error. When researchers reject the null hypothesis and it is true, they have made a Type I error. We can use the $z$-observed value and Appendix Al to calculate the probability of Type I error if the null hypothesis were perfectly true and the researcher chose to reject the null hypothesis (regardless of the $\alpha$ level). This probability is called a $p$-value (abbreviated as $p$ ). Visually, it can be represented, as in Figure $7.6$, as the region of the sampling distribution that starts at the observed value and includes everything beyond it in the tail.

To calculate $p$, you should first find the $z$-observed value in column $\mathrm{A}$. (If the $z$-observed value is not in Appendix A1, then the Price Is Right rule applies, and you should select the number in column A that is closest to the $z$-observed value without going over it.) The number in column $\mathrm{C}$ in the same row will be the $p$-value. For example, in a one-tailed test, if $z$-observed were equal to $+2.10$, then the $p$-value (i.e., the number in column $\mathrm{C}$ in the same row) would be 0179 . This $p$-value means that in this example the probability that these results could occur through purely random sampling error is .0179 (or $1.79 \%$ ). In other words, if we selected an infinite number of samples from the population, then $1.79 \%$ of $\bar{X}$ values would be as different as or more different than $\mu$. But remember that this probability only applies if the null hypothesis is perfectly true (see Sidebar 10.3).

In the Vinten et al. $(2009)$ example, the $z$-observed value was $-9.95$. However, because Appendix Al does not have values that high, we will select the last row (because of the Price Is Right rule), which has the number $\pm 5.00$ in column A. The number in column $\mathrm{C}$ in the same row is $.0000003$, which is the closest value available for the $p$-value. In reality, $p$ will be smaller than this tiny number. (Notice how the numbers in column $C$ get smaller as the numbers in column $A$ get bigger. Therefore, a $z$-observed value that is outside the $\pm 5.00$ range will have smaller $p$-values than the values in the table.) Thus, the chance that – if the null hypothesis were true – Vinten et al. (2009) would obtain a random sample of 41 children with such low VABS scores is less than .0000003-or less than 3 in 10 million. Given this tiny probability of making a Type I error if the null hypothesis were true, it seems more plausible that these results are due to an actual difference between the sample and the population – and not merely to sampling error.

## 统计代写|Generalized linear model代考广义线性模型代写|Null Hypothesis Statistical Significance Testing

NHST 程序有八个步骤：

1. 在数据中形成组。
2. 定义零假设(H0). 零假设始终是组之间没有差异，或者自变量和因变量之间没有关系。
3. 设置阿尔法(一种). 默认阿尔法=.05.
4. 选择单尾或双尾测试。这决定了备择假设(Hl).
5. 找到用于定义拒绝区域的临界值。计算观察值。
6. 比较观察值和临界值。如果观测值比临界值更极端，则应拒绝原假设。否则，应保留。
7. 计算效应量。
8. 以前没有接触过统计数据的读者现在会发现这些步骤令人困惑和抽象。但本章的其余部分将定义术语并展示如何将这些步骤付诸实践。为了减少混淆，本书从最简单的 NHST 开始：和-测试。

## 统计代写|Generalized linear model代考广义线性模型代写|Cautions for Using NHSTs

NHST 程序主导了行为科学的定量研究（Cumming 等人，2007；Fidler 等人，2005；Warne，Lazo，Ramos，\& Ritter，2012）。但这不是一个完美的程序，NHST 很容易被滥用。在本节中，我们将探讨 NHST 的三个主要问题：(1) 错误的可能性，(2) 进行 NHST 所涉及的主观决定，以及 (3) NHST 对样本量的敏感性。

I 型和 II 型错误。在文滕等人。（2009）的例子，我们拒绝了原假设，因为和- 观察值在拒绝区域内，如图所示7.4（其中 z 观测值远低于零，无法在图中显示）。但这并不意味着零假设肯定是错误的。请记住，理论上——概率分布从−∞到 到 。因此，一个随机样本可能有一个X¯值与 Vinten 等人 (2009) 研究中观察到的值一样低。这显然是一个不太可能发生的事件——但从理论上讲，这是可能的。所以即使我们拒绝了原假设和X¯这个例子中的值有一个非常极端的 z 观察值，Vinten 等人仍然有可能。只是有一个奇怪的样本（这会产生大量的抽样误差）。因此，该 z 检验的结果并不能证明抗癫痫药物对子宫内的儿童有害。

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