Source of data: Golub et al$.(1999).$ Molecular classification of cancer: class discovery and class prediction by

gene expression monitoring, Science, Vol. $286$:$531-537.$

The data set golub consists of the expression levels of $3051$ genes for $38$ tumor mRNA samples. Each tumor

mRNA sample comes from one patient $($i$.$e$.38$ patients total$),$ and $27$ of these tumor samples correspond to

acute lymphoblastic leukemia $($ALL$)$ and the remaining $11$ to acute myeloid leukemia $($AML$).$

You will need to discover how many genes can be used to differentiate the tumor types $($meaning that their

expression level differs between the two tumor types$)$ using

the uncorrected p$-$values,

the Holm$-$Bonferroni correction, and $($iii$)$ the Benjamini$-$Hochberg correction?

Feel free to use libraries for multiple hypothesis testing in $R$ or python.

If you are using Python, you can use the following code to load the data:

with zipfile.ZipFile$("$statsreview$\_$release$1.$zip"$)$ as zip$\_$file:

golub$\_$data, golub$\_$classnames $=($ np$.$genfromtxt$($zip$\_$file.open$(\text{'}$data$\_$and$\_$materials$/$

golub$\_$data$/\{\}\text{'}.$format$($fname$)),$ delimiter$=\text{'},\text{'},$ names$=$True, converters$=\{0$: lambda s:

int$($s$.$strip$($b$\text{'}"\text{'}))\})$ for fname in $[\text{'}$golub$.$csve, 'golub$\_$cl$.$csv$\text{'}])$

Part $($a$)$

$\frac{0\mathrm{.}0}{2\mathrm{.}0}$ points $($graded$)$

Let ${\stackrel{}{x}}_{\text{A}\text{L}\text{L}\text{,}i}$ be the mean of the expression levels for gene i across the ALL mRNA samples. Similarly, let

${\stackrel{}{x}}_{\text{A}\text{M}\text{L}\text{,}i}$ be the same but for the AML mRNA samples instead.

For each of these, $N_{\text{A}\text{L}\text{L}}$ and $N_{\text{A}\text{M}\text{L}}$ are the number of mRNA samples for the ALL tumors and AML tumors

respectively.

If $s_{\text{A}\text{L}\text{L}\text{,}i}^2$ is the sample variance for gene i across the ALL mRNA observations, then the corresponding

variance for ${\stackrel{}{x}}_{\text{A}\text{L}\text{L}\text{,}i}$ is

$s_{{\stackrel{}{x}}_{\text{A}\text{M}\text{L}\text{,}}^2}^2=\frac{s_{\text{A}\text{M}\text{L}\text{,}i}^2}{N_{\text{A}\text{M}\text{L}}}$

We can use ${\stackrel{}{x}}_i={\stackrel{}{x}}_{\text{A}\text{L}\text{L}\text{,}i}-{\stackrel{}{x}}_{\text{A}\text{M}\text{L}\text{,}i}$ as a metric for the difference in expression levels for gene $i.$ The

variance of this metric is

$s_{{\stackrel{}{x}}_i}^2=s_{{\stackrel{}{x}}_{\text{A}\text{L}\text{L}},i}^2+s_{{\stackrel{}{x}}_{\text{A}\text{M}\text{L}},i}^2$

This allows us to use the following test statistic:

$t_{\text{W}\text{e}\text{l}\text{c}\text{h}\text{,}i}=\frac{{\stackrel{}{x}}_{\text{A}\text{L}\text{L}\text{,}i}-{\stackrel{}{x}}_{\text{A}\text{M}\text{L}\text{,}i}}{\sqrt[\phantom{2}]{\frac{s_{\text{A}\text{L}\text{L}\text{,}i}^2}{N_{\text{A}\text{L}\text{L}}}+\frac{s_{\text{A}\text{M}\text{L}\text{,}i}^2}{N_{\text{A}\text{M}\text{L}}}}}$

which you can recognize as similar to the t$-$test statistic, and is itself known as the Welch unequal variances t$-$

test.

The distribution for the Welch test statistic can be approximated by a t$-$distribution, but with a modified

number of degrees of freedom. The number of degrees of freedom is approximately

$)i)ALL)AML$

where $)ALL$ and $)AML.$

Use the Welch t$-$test to find the number of significantly associated genes $(0\mathrm{.}05)$ using uncorrected p$-$

values.

How many genes are significant? $($Please enter the value with a precision of at least two significant figures,

your answer will be graded with a $10\%$ tolerance.$)$