{"id":3786,"date":"2024-05-27T00:44:15","date_gmt":"2024-05-26T15:44:15","guid":{"rendered":"https:\/\/bdsl.jbnu.ac.kr\/blog\/?p=3786"},"modified":"2024-05-27T00:56:36","modified_gmt":"2024-05-26T15:56:36","slug":"understanding-the-concept-of-statistics-through-examples","status":"publish","type":"post","link":"https:\/\/bdsl.jbnu.ac.kr\/blog\/understanding-the-concept-of-statistics-through-examples\/","title":{"rendered":"Understanding the Concept of Statistics Through Examples"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Library loading and functions<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Python libraries and functions are loaded before the tutorial.<\/li>\n<\/ul>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">from scipy import stats\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\ndef generate_random_values_A(n_sample=5):\n    rna = stats.norm.rvs(loc=10, scale=2, size=n_sample)\n    protein = rna * 2 + stats.norm.rvs(loc=0, scale= 1, size=n_sample) * 3\n    return (rna, protein)\n\ndef generate_random_values_B(n_sample=5):\n    rna = stats.norm.rvs(loc=10, scale=2, size=n_sample)\n    protein = stats.norm.rvs(loc=10, scale=2, size=n_sample) * 2\n    return (rna, protein)\n\ndef plot_relationship(input):\n    (rna, protein) = input\n    fig = plt.figure(figsize=(4, 4))\n    \n    ax = fig.add_subplot(111)\n    ax.scatter(rna, protein, zorder=2)\n    ax.set_xlabel('RNA concentration (A)')\n    ax.set_ylabel('Protein concentration (A)')\n    ax.grid(zorder=1)\n\ndef calculate_correlation(input):\n    (rna, protein) = input\n    rho, pv = stats.pearsonr(rna, protein)\n\n    return rho, pv<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">RNA expression and protein expression<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Suppose that we want to determine whether the RNA and protein expression levels for Protein A are correlated.\n<ul class=\"wp-block-list\">\n<li>Correlated: showing a relationship between each other.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Hypothesis: The RNA and Protein expression levels of Protein A are correlated.<\/li>\n\n\n\n<li>Experiment: Conduct experiments to measure both RNA and protein concentrations.<\/li>\n\n\n\n<li>Data analysis: Use statistical methods to analyze the data. We will explore this here.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Measure of the concentration of the protein X<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Suppose we measured the RNA and protein concentration of Protein A in five samples.<\/li>\n\n\n\n<li>The values are plotted below.<\/li>\n<\/ul>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\"># Perform experiments and get the results\nrvs = generate_random_values_A()\n\n# Plot the results\nplot_relationship(rvs)<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"377\" height=\"371\" src=\"https:\/\/bdsl.jbnu.ac.kr\/blog\/wp-content\/uploads\/2024\/05\/image.png\" alt=\"\" class=\"wp-image-3787\" srcset=\"https:\/\/bdsl.jbnu.ac.kr\/blog\/wp-content\/uploads\/2024\/05\/image.png 377w, https:\/\/bdsl.jbnu.ac.kr\/blog\/wp-content\/uploads\/2024\/05\/image-300x295.png 300w\" sizes=\"auto, (max-width: 377px) 100vw, 377px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Interepretation<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Now that we have the data, how can we interpret it to determine whether the RNA and protein concentrations are correlated?<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">A Statistic: A Single Number that Represents the Data<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We can devise a measure to evaluate relationship:\n<ul class=\"wp-block-list\">\n<li>A statistic called Pearson&#8217;s correlation coefficient.<\/li>\n\n\n\n<li>The value would be 0 if the two variables, x and y are not correlated.<\/li>\n\n\n\n<li>The value would be &gt; 0 if x and y change together.<\/li>\n\n\n\n<li>The value would be &lt; 0 if x and y changes in opposite directions.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>The formula for Peason&#8217;s correlation coefficient is<\/li>\n\n\n\n<li>\\(r_{xy} = \\frac{\\sum_{i=1}^n{(x_i &#8211; \\overline{x})(y_i &#8211; \\overline{y})}}{\\sqrt{\\sum_{i=1}^n{(x_i &#8211; \\overline{x})^2}}\\sqrt{\\sum_{i=1}^n{(y_i &#8211; \\overline{y})^2}}}\\)\u00a0\u00a0<\/li>\n<\/ul>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">calculate_correlation(rvs)<\/code><\/pre>\n\n\n\n<ul class=\"wp-block-list\">\n<li>PCC = 0.83<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Simple Evaluation<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We have a statistic called Pearson&#8217;s correlation coefficient, which has a value of 0.86.<\/li>\n\n\n\n<li>Since the value is larger than 0, we can say that the experimental results support the hypothesis that the RNA and protein expression levels of Protein A are positively correlated.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Random Incidences<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>However, we need to consider whether our interpretation is valid.<\/li>\n\n\n\n<li>To test this, we will generate the same statistic using two independent variables, repeating this process five times.<\/li>\n\n\n\n<li>We found 3 out of 5 random association had a Pearson correlation coefficient (PCC) greater than 0.<\/li>\n\n\n\n<li>Therefore, observing a PCC greater than 0 might be too loose a criteria.<\/li>\n<\/ul>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">for i in range(5):\n    rvs = generate_random_values_B()\n    print(calculate_correlation(rvs)[0])   <\/code><\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">0.5148158483005283<br>-0.6822188844235954<br>0.43813562081717283<br>-0.2495326763997526<br>0.10300182994047037<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">Statistical Inference<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Let&#8217;s generate more random numbers and calculate PCC multiple times.<\/li>\n\n\n\n<li>We repated this process 1000 times, obtaining a distribution of PCC values.<\/li>\n\n\n\n<li>Among these, 30 instances had a PCC greater than 0.85, representing 3% of the cases (30\/1000 = 3%).<\/li>\n\n\n\n<li>In other words, we can observer a PCC larger than 0.85 in 3% of the experiments, even when the two variables are not correlated.<\/li>\n<\/ul>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">rhos= []\ncnt = 0 \nfor i in range(1000):\n    rvs = generate_random_values_B()\n    rhos.append(calculate_correlation(rvs)[0]) \n    if calculate_correlation(rvs)[0] &gt; 0.85:\n        cnt += 1\n\nsns.histplot(rhos)\nprint(cnt)\n<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"571\" height=\"413\" src=\"https:\/\/bdsl.jbnu.ac.kr\/blog\/wp-content\/uploads\/2024\/05\/image-1.png\" alt=\"\" class=\"wp-image-3788\" srcset=\"https:\/\/bdsl.jbnu.ac.kr\/blog\/wp-content\/uploads\/2024\/05\/image-1.png 571w, https:\/\/bdsl.jbnu.ac.kr\/blog\/wp-content\/uploads\/2024\/05\/image-1-300x217.png 300w\" sizes=\"auto, (max-width: 571px) 100vw, 571px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">P-value<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We can calculate the probability of observing a statistic when the two variables are not correlated.<\/li>\n\n\n\n<li>The probability is called the p-value.<\/li>\n\n\n\n<li>We can estimate the p-value through simulation, as demonstrated above.<\/li>\n\n\n\n<li>However, in certain cases, we can calculate the p-value exactly.<\/li>\n\n\n\n<li>In other words, p-values can often be calculated quickly without the need for simulation.<\/li>\n\n\n\n<li>Pearson&#8217;s correlation coefficient is one such statistic with a well defined distribution, allowing us to calculate the p-value accurately.<\/li>\n\n\n\n<li>You can calculate PCC using <code>stats.pearsonr<\/code> function of <code>scipy<\/code> package.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Statistical inference<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Now, we can make a statistical inference using the p-value.<\/li>\n\n\n\n<li>Typically, we set a cutoff for the p-value, with 0.05 or 5% being the most commonly used threshold.<\/li>\n\n\n\n<li>If the p-value of a statisic for an observation is less than 5%, we call the observation statistically significant, as it is unlikely (less than 5% probability) to have occurred by chance.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Library loading and functions RNA expression and protein expression Measure of the concentration of the protein X Interepretation A Statistic: A Single Number that Represents the Data Simple Evaluation Random Incidences 0.5148158483005283-0.68221888442359540.43813562081717283-0.24953267639975260.10300182994047037 Statistical Inference P-value Statistical inference<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center 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