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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
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## 统计代写|生物统计分析代写Biological statistic analysis代考|Effect Size Measures

The raw difference $\Delta$ or the standardized difference $d$ are both easily interpretable effect size measures for the case of $k=2$ treatment groups that we can use in conjunction with the $t$-test. We now introduce three effect size measures for the case of $k>2$ treatment groups for use in conjunction with an omnibus $F$-test.

A simple effect size measure is the variation explained, which is the proportion of the factor’s sum of squares of the total sum of squares:
$$\eta_{\mathrm{trt}}^2=\frac{\mathrm{SS}{\mathrm{trt}}}{\mathrm{SS}{\mathrm{tot}}}=\frac{\mathrm{SS}{\mathrm{trt}}}{\mathrm{SS}{\mathrm{trt}}+\mathrm{SS}{\mathrm{res}}} .$$ A large value of $\eta{\text {trt }}^2$ indicates that the majority of the variation is not due to random variation between observations in the same treatment group, but rather due to the fact that the average responses to the treatments are different. In our example, we find $\eta_{\mathrm{trt}}^2=155.89 / 197.26=0.79$, confirming that the differences between drugs are responsible for $79 \%$ of the variation in the data.

The raw effect size measures the average deviation between group means and the grand mean:
$$b^2=\frac{1}{k} \sum_{i=1}^k\left(\mu_i-\mu\right)^2=\frac{1}{k} \sum_{i=1}^k \alpha_i^2 .$$
A corresponding standardized effect size was proposed in Cohen (1992):
$$f^2=b^2 / \sigma_e^2=\frac{1}{k \sigma_e^2} \sum_{i=1}^k \alpha_i^2$$
and measures the average deviation between group means and grand mean in units of residual variance. It specializes to $f=d / 2$ for $k=2$ groups.

Extending from his classification of effect sizes $d$, Cohen proposed that values of $f>0.1, f>0.25$, and $f>0.4$ may be interpreted as small, medium, and large effects (Cohen 1992). An unbiased estimate of $f^2$ is
$$\hat{f}^2=\frac{\mathrm{SS}{\mathrm{trt}}}{\mathrm{SS}{\mathrm{res}}}=\frac{k-1}{N-k} \cdot F=\frac{N}{N-k} \cdot \frac{1}{k \cdot \hat{\sigma}e^2} \sum{i=1}^k \hat{\alpha}_i^2,$$
where $F$ is the observed value of the $F$-statistic, yielding $\hat{f}^2=3.77$ for our example; the factor $N /(N-k)$ removes the bias.

## 统计代写|生物统计分析代写Biological statistic analysis代考|General Idea

Recall that under the null hypothesis of equal treatment group means, the deviations $\alpha_i=\mu_i-\mu$ are all zero, and $F=\mathrm{MS}{\mathrm{tr}} / \mathrm{MS}{\text {res }}$ follows an $F$-distribution with $k-1$ numerator and $N-k$ denominator degrees of freedom.

If the treatment effects are not zero, then the test statistic follows a noncentral F-distribution $F_{k-1, N-k}(\lambda)$ with noncentrality parameter
$$\lambda=n \cdot k \cdot f^2=\frac{n \cdot k \cdot b^2}{\sigma_e^2}=\frac{n}{\sigma_e^2} \cdot \sum_{i=1}^k \alpha_i^2 .$$
The noncentrality parameter is thus the product of the overall sample size $N=n \cdot k$ and the standardized effect size $f^2$ (which can be translated from and to $\eta_{\text {trt }}^2$ via Eq. (4.1)). For $k=2$, this reduces to the previous case since $f^2=d^2 / 4$ and $t_n^2(\eta)=$ $F_{1, n}\left(\lambda=\eta^2\right)$.

The idea behind the power analysis is the same as for the $t$-test. If the omnibus hypothesis $H_0$ is true, then the $F$-statistic follows a central $F$-distribution with $k-1$ and $N-k$ degrees of freedom, shown in Fig. $4.2$ (top) for two sample sizes $n=2$ (left) and $n=10$ (right). The hypothesis is rejected at the significance level $\alpha=5 \%$ (black shaded area) whenever the observed $F$-statistic is larger than the $95 \%$ quantile $F_{1-\alpha, k-1, N-k}$ (dashed line).

If $H_0$ is false, then the $F$-statistic follows a noncentral $F$-distribution. Two corresponding examples are shown in Fig.4.2 (bottom) for an effect size $f^2=0.34$ corresponding. for example, to a difference of $\delta=2$ between the first and the second treatment group, with no difference between the remaining two treatment groups. We observe that this distribution shifts to higher values with increasing sample size, since its noncentrality parameter $\lambda=n \cdot k \cdot f^2$ increases with $n$. For $n=2$, we have $\lambda=2 \cdot 4 \cdot f^2=2.72$ in our example, while for $n=10$, we already have $\lambda=10 \cdot 4 \cdot f^2=13.6$. In each case, the probability $\beta$ of falsely not rejecting $H_0$ (a false positive) is the gray shaded area under the density up to the rejection quantile $F_{1-\alpha, k-1, N-k}$ of the central $F$-distribution. For $n=2$, the corresponding power is then $1-\beta=13 \%$ which increases to $1-\beta=85 \%$ for $n=10$.

# 生物统计代考

## 统计代写|生物统计分析代写Biological statistic analysis代考|效应量测量

$$\eta_{\text {trt }}^2=\frac{\text { SStrt }}{\text { SStot }}=\frac{\text { SStrt }}{\text { SStrt }+\text { SSres }} .$$

$$b^2=\frac{1}{k} \sum_{i=1}^k\left(\mu_i-\mu\right)^2=\frac{1}{k} \sum_{i=1}^k \alpha_i^2$$
Cohen (1992) 提出了相应的标准化效应量:
$$f^2=b^2 / \sigma_e^2=\frac{1}{k \sigma_e^2} \sum_{i=1}^k \alpha_i^2$$

$$\hat{f}^2=\frac{\text { SStrt }}{\text { SSres }}=\frac{k-1}{N-k} \cdot F=\frac{N}{N-k} \cdot \frac{1}{k \cdot \hat{\sigma} e^2} \sum i=1^k \hat{\alpha}_i^2,$$

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