### 统计代写|统计推断作业代写statistics interference代考|Significance tests

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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|统计推断作业代写statistics interference代考|General remarks

So far, in our frequentist discussion we have summarized information about the unknown parameter $\psi$ by finding procedures that would give in hypothetical repeated applications upper (or lower) bounds for $\psi$ a specified proportion of times in a long run of repeated applications. This is close to but not the same as specifying a probability distribution for $\psi$; it avoids having to treat $\psi$ as a random variable, and moreover as one with a known distribution in the absence of the data.

Suppose now there is specified a particular value $\psi_{0}$ of the parameter of interest and we wish to assess the relation of the data to that value. Often the hypothesis that $\psi=\psi_{0}$ is called the null hypothesis and conventionally denoted by $H_{0}$. It may, for example, assert that some effect is zero or takes on a value given by a theory or by previous studies, although $\psi_{0}$ does not have to be restricted in that way.

There are at least six different situations in which this may arise, namely the following.

• There may be some special reason for thinking that the null hypothesis may be exactly or approximately true or strong subject-matter interest may focus on establishing that it is likely to be false.
• There may be no special reason for thinking that the null hypothesis is true but it is important because it divides the parameter space into two (or more) regions with very different interpretations. We are then interested in whether the data establish reasonably clearly which region is correct, for example it may establish the value of $\operatorname{sgn}\left(\psi-\psi_{0}\right)$.
• Testing may be a technical device used in the process of generating confidence intervals.
• Consistency with $\psi=\psi_{0}$ may provide a reasoned justification for simplifying an otherwise rather complicated model into one that is more transparent and which, initially at least, may be a reasonable basis for interpretation.
• Only the model when $\psi=\psi_{0}$ is under consideration as a possible model for interpreting the data and it has been embedded in a richer family just to provide a qualitative basis for assessing departure from the model.
• Only a single model is defined, but there is a qualitative idea of the kinds of departure that are of potential subject-matter interest.

The last two formulations are appropriate in particular for examining model adequacy.

From time to time in the discussion it is useful to use the short-hand description of $H_{0}$ as being possibly true. Now in statistical terms $H_{0}$ refers to a probability model and the very word ‘model’ implies idealization. With a very few possible exceptions it would be absurd to think that a mathematical model is an exact representation of a real system and in that sense all $H_{0}$ are defined within a system which is untrue. We use the term to mean that in the current state of knowledge it is reasonable to proceed as if the hypothesis is true. Note that an underlying subject-matter hypothesis such as that a certain environmental exposure has absolutely no effect on a particular disease outcome might indeed be true.

## 统计代写|统计推断作业代写statistics interference代考|Simple significance test

In the formulation of a simple significance test, we suppose available data $y$ and a null hypothesis $H_{0}$ that specifies the distribution of the corresponding random variable $Y$. In the first place, no other probabilistic specification is involved, although some notion of the type of departure from $H_{0}$ that is of subject-matter concern is essential.

The first step in testing $H_{0}$ is to find a distribution for observed random variables that has a form which, under $H_{0}$, is free of nuisance parameters, i.e., is

completely known. This is trivial when there is a single unknown parameter whose value is precisely specified by the null hypothesis. Next find or determine a test statistic $T$, large (or extreme) values of which indicate a departure from the null hypothesis of subject-matter interest. Then if $t_{\text {obs }}$ is the observed value of $T$ we define
$$p_{\mathrm{obs}}=P\left(T \geq t_{\mathrm{obs}}\right) \text {, }$$
the probability being evaluated under $H_{0}$, to be the (observed) $p$-value of the test.

It is conventional in many fields to report only very approximate values of $p_{\text {obs }}$, for example that the departure from $H_{0}$ is significant just past the 1 per cent level, etc.

The hypothetical frequency interpretation of such reported significance levels is as follows. If we were to accept the available data as just decisive evidence against $H_{0}$, then we would reject the hypothesis when true a long-run proportion pobs of times.

Put more qualitatively, we examine consistency with $H_{0}$ by finding the consequences of $H_{0}$, in this case a random variable with a known distribution, and seeing whether the prediction about its observed value is reasonably well fulfilled.

We deal first with a very special case involving testing a null hypothesis that might be true to a close approximation.

## 统计代写|统计推断作业代写statistics interference代考|One- and two-sided tests

In many situations observed values of the test statistic in either tail of its distribution represent interpretable, although typically different, departures from $H_{0}$ The simplest procedure is then often to contemplate two tests, one for each tail, in effect taking the more significant, i.e., the smaller tail, as the basis for possible interpretation. Operational interpretation of the result as a hypothetical error rate is achieved by doubling the corresponding $p$, with a slightly more complicated argument in the discrete case.

More explicitly we argue as follows. With test statistic $T$, consider two $p$-values, namely
$$p_{\mathrm{obs}}^{+}=P\left(T \geq t ; H_{0}\right), \quad p_{\mathrm{obs}}^{-}=P\left(T \leq t ; H_{0}\right) .$$
In general the sum of these values is $1+P(T=t)$. In the two-sided case it is then reasonable to define a new test statistic
$$Q=\min \left(P_{\mathrm{obs}}^{+}, P_{\text {obs }}^{-}\right)$$
The level of significance is
$$P\left(Q \leq q_{\text {obs }} ; H_{0}\right)$$
In the continuous case this is $2 q_{\text {obs }}$ because two disjoint events are involved. In a discrete problem it is $q_{\text {obs plus the achievable } p \text {-value from the other tail }}$ of the distribution nearest to but not exceeding $q_{\mathrm{obs}}$. As has been stressed the precise calculation of levels of significance is rarely if ever critical, so that the careful definition is more one of principle than of pressing applied importance. A more important point is that the definition is unaffected by a monotone transformation of $T$.

In one sense very many applications of tests are essentially two-sided in that, even though initial interest may be in departures in one direction, it will rarely be wise to disregard totally departures in the other direction, even if initially they are unexpected. The interpretation of differences in the two directions may well be very different. Thus in the broad class of procedures associated with the linear model of Example $1.4$ tests are sometimes based on the ratio of an estimated variance, expected to be large if real systematic effects are present, to an estimate essentially of error. A large ratio indicates the presence of systematic effects whereas a suspiciously small ratio suggests an inadequately specified model structure.

## 统计代写|统计推断作业代写statistics interference代考|General remarks

• 可能有一些特殊的理由认为零假设可能完全或近似正确，或者强烈的主题兴趣可能集中在确定它可能是错误的。
• 认为零假设为真可能没有特别的理由，但它很重要，因为它将参数空间划分为两个（或更多）具有非常不同解释的区域。然后，我们对数据是否合理清楚地确定哪个区域是正确的感兴趣，例如它可以确定sgn⁡(ψ−ψ0).
• 测试可能是在生成置信区间的过程中使用的技术设备。
• 一致性ψ=ψ0可以为将原本相当复杂的模型简化为更透明的模型提供合理的理由，并且至少在最初可能是解释的合理基础。
• 仅当模型ψ=ψ0正在考虑作为解释数据的可能模型，它已被嵌入到一个更丰富的家庭中，只是为了提供一个定性基础来评估偏离模型的情况。
• 仅定义了一个模型，但对具有潜在主题兴趣的偏离类型有一个定性的想法。

p这bs=磷(吨≥吨这bs),

## 统计代写|统计推断作业代写statistics interference代考|One- and two-sided tests

p这bs+=磷(吨≥吨;H0),p这bs−=磷(吨≤吨;H0).

## 广义线性模型代考

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## MATLAB代写

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