# 数学代写|基础数据分析代写Elementary data Analysis代考|MATH1105

## 数学代写|基础数据分析代写Elementary data Analysis代考|On Significant Coefficients

If all the usual distributional assumptions hold, then $t$-tests can be used to decide whether particular coefficients are statistically-significantly different from zero. Pretty much any piece of statistical software, $R$ very much included, reports the results of these tests automatically. It is far too common to seriously over-interpret those results, for a variety of reasons.

Begin with what hypothesis, exactly, is being tested when $R$ (or whatever) runs those $t$-tests. Say, without loss of generality, that there are $p$ predictor variables, $\vec{X}=\left(X_1, \ldots X_p\right)$, and that we are testing the coefficient on $X_p$. Then the null hypothesis is not just ” $\beta_p=0$ “, but ” $\beta_p=0$ in a linear model which also includes $X_1, \ldots X_{p-1}$, and nothing else”. The alternative hypothesis is not just “ $\beta_p \neq 0$ “, but “ $\beta_p \neq 0$ in a model which also includes $X_1, \ldots X_{p-1}$, but nothing else”. The optimal linear coefficient on $X_p$ will depend not just on the relationship between $X_p$ and the response $Y$, but also on which other variables are included in the model. The $t$-test checks whether adding $X_p$ really improves predictions more than would be expected, under all these assumptions, if one is already using all the other variables, and only those other variables. It does not and cannot test whether $X_p$ is important in any absolute sense.

Even if you are willing to say “Yes, all I really want to know about this variable is whether adding it to the model really helps me predict in a linear approximation”, remember that the question which a $t$-test answers is whether adding that variable will help at all. Of course, as you know from your regression class, and as we’ll see in more detail in Chapter 3, expanding the model never hurts its performance on the training data. The point of the $t$-test is to gauge whether the improvement in prediction is small enough to be due to chance, or so large, compared to what noise could produce, that one could confidently say the variable adds some predictive ability. This has several implications which are insufficiently appreciated among users.

In the first place, tests on individual coefficients can seem to contradict tests on groups of coefficients. Adding multiple variables to the model could significantly improve the fit (as checked by, say, a partial $F$ test), even if none of the coefficients is significant on its own. In fact, every single coefficient in the model could be insignificant, while the model as a whole is highly significant (i.e., better than a flat line).

## 数学代写|基础数据分析代写Elementary data Analysis代考|Error and Inference

There are (at least) three ways we can use statistical models in data analysis: as summaries of the data, as predictors, and as simulators.

The least demanding use of a model is to summarize the data – to use it for data reduction, or compression. Just as the sample mean or sample quantiles can be descriptive statistics, recording some features of the data and saying nothing about a population or a generative process, we could use estimates of a model’s parameters as descriptive summaries. Rather than remembering all the points on a scatter-plot, say, we’d just remember what the OLS regression surface was.

It’s hard to be wrong about a summary, unless we just make a mistake. (It may not be helpful for us later, but that’s different.) When we say “the slope which minimized the sum of squares was $4.02^{\prime \prime}$, we make no claims about anything but the training data. That statement relies on no assumptions, beyond our calculating correctly. But it also asserts nothing about the rest of the world. As soon as we try to connect our training data to anything else, we start relying on assumptions, and we run the risk of being wrong.

Probably the most common connection to want to make is to say what other data will look like – to make predictions. In a statistical model, with random variables, we do not anticipate that our predictions will ever be exactly right, but we also anticipate that our mistakes will show stable probabilistic patterns. We can evaluate predictions based on those patterns of error $-$ how big is our typical mistake? are we biased in a particular direction? do we make a lot of little errors or a few huge ones?
Statistical inference about model parameters – estimation and hypothesis testing – can be seen as a kind of prediction, extrapolating from what we saw in a small piece of data to what we would see in the whole population, or whole process.

# 基础数据分析代考

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## 数学代写|基础数据分析代写基本数据分析代考|错误和推断

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