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Updated page "look" using quarto.

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.quarto/_freeze/F_test/execute-results/html.json

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+    "markdown": "---\ntitle: \"Comparing variances: Fisher's F-test\"\neditor_options: \n  chunk_output_type: console\n---\n\n\n*Last modified on 2022-09-12*\n\n\n::: {.cell}\n\n:::\n\n::: {.cell}\n\n:::\n\n\n\n\nThe test of variances requires that the two sampled population be normally distributed and that the samples are randomly selected from their respective populations.\n\n# Introduction\n\nThe method is simple; it consists of taking the ratio between the larger population variance, $\\sigma_1^2$, and the smaller population variance, $\\sigma_2^2$, then looking up the ratio on an $F$-distribution curve. The null hypothesis states that the ratio equal 1,\n\n$$\nH_o: \\frac{\\sigma_1^2}{\\sigma_2^2} = 1\n$$\n\nand the alternate hypothesis states that the ratio differs from 1 (i.e. the variances differ),\n\n$$\nH_a: \\frac{\\sigma_1^2}{\\sigma_2^2} \\neq  1\n$$\n\nor is greater than 1 (i.e. $\\sigma_1^2$ is significantly bigger than $\\sigma_2^2$),\n\n$$\nH_a: \\frac{\\sigma_1^2}{\\sigma_2^2} \\gt  1\n$$\n\n\nSince the larger variance is assigned to the numerator by convention, we do not have a situation where the ratio is less than 1.\n\nSince we are working with samples, we work with the sample variances $s_1^2$ and $s_2^2$ and compute the test statistic $F$ as follows:\n\n$$\nF = \\frac{s_1^2}{s_2^2}\n$$\n\n\nThe shape of the $\\pmb F$**-distribution** curve is defined by both sample's $df$'s, i.e. $(n_1-1)$ and $(n_2-1)$. Like the $\\chi^2$ distribution, the $F$ distribution tends to be skewed to the right, especially for large $df$'s.\n\n# Example 1\n\nIn one of the examples in the _$z$ and $t$ test_ [tutorial](z_t_tests.html#example-1), we seek to compare the concentration of sulfates between background sites and a contaminated well (data taken from Millard _et al._, p. 418). Did the two samples have equal variances? The table of concentrations is reproduced here.\n\nContaminated      Background\n---------------   -------------------\n600               560\n590               550\n570               570\n570               550\n565               570\n580               590\n                  550\n                  580\n\n## Solution to example 1\n\nThe variances for both samples are $s_{Ref}^2 = 712.5$ and $s_{Cont}^2 = 336.7$. Since $s_{Ref}^2 > s_{Cont}^2$, the value $s_{Ref}^2$ will be in the numerator giving us the following test statistic:\n\n\n$$\nF = \\frac{s_{Ref}^2}{s_{Cont}^2} = \\frac{712.5}{336.7} = 2.12\n$$\n\n\nNext, we must determine where the $F$ statistic lies along the $F$-distribution curve. This requires that we compute the two $df$'s from the samples to define the shape of the $F$ distribution:\n\n$$\ndf_{Ref} = 8 - 1 = 7\n$$\n\n$$\ndf_{Cont} = 6 - 1 =5\n$$\n\n\nNow that we have the shape of the $F$-distribution defined, we can look up the probability of getting an $F$ statistic as extreme as ours, An F-distribution table can be used, or the value can be computed exactly using the function `pf()`:\n\n\n::: {.cell}\n\n```{.r .cell-code}\npf(2.12, 7, 5, lower.tail=FALSE)\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n[1] 0.2126279\n```\n:::\n:::\n\n::: {.cell}\n::: {.cell-output-display}\n![](F_test_files/figure-html/unnamed-chunk-3-1.png){width=672}\n:::\n:::\n\n\nThe $F$ values associated with a probability of 0.025 and 0.975 (associated with rejection regions for a two-tailed $\\alpha$ of 0.05) are displayed on the curve in grey dashed vertical lines.\n\nThe probability of getting an $F$ as large as ours is about 0.21 (or 21%). Since $H_a$ represents _both_ sides of the distribution, we double the probability to give us the chance of getting a test statistic as great or as small as ours, so for a two-tailed test, $P=0.42$. With such a high $P$-value, we cannot reject the null and therefore can state that for all intents and purposes, the variances between both populations are the same (i.e. the observed variability between both $s$ can be explain by chance alone).\n\nThe following figure shows the observed $P$ values in both tails.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![](F_test_files/figure-html/unnamed-chunk-4-1.png){width=672}\n:::\n:::\n\n\nThis can be easily executed in `R` as a two-tailed test as shown in the following code block:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nRef <-  c(560, 530, 570, 490, 510, 550, 550, 530)\nCont <- c(600, 590, 590, 630, 610, 630)\nvar.test(Ref, Cont, alternative=\"two.sided\")\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n\n\tF test to compare two variances\n\ndata:  Ref and Cont\nF = 2.1163, num df = 7, denom df = 5, p-value = 0.4263\nalternative hypothesis: true ratio of variances is not equal to 1\n95 percent confidence interval:\n  0.3088156 11.1853404\nsample estimates:\nratio of variances \n          2.116337 \n```\n:::\n:::\n\n\nNote that the `var.test()` computes the $F$ ratio using the first variable name in the list as the numerator. For example, had we reversed the order of variables (i.e. `var.test(Cont, Ref, alternative=\"two-sided\")`), the returned $F$ value would be the inverse  of the original $F$ value, or $1/2.12 = 0.47$. The $P$ value would have stayed the same however.\n\n# Example 2\n\nAn investor is concerned that stock 1 is a riskier investment than stock 2 because its variation in daily prices is greater.  The following table provides summary statistics for a sample of 25 daily price changes. \n\n                     Stock 1   Stock 2\n------------------- --------- --------\nSample size          25        25\nStandard deviation   .76       .46\n\nIs stock 1's variability significantly greater than that of stock 2, or is the observed difference due to chance?  \n_[This example is adapted from McCLave et al. page 461]_\n\n## Solution to example 2\n\nWe are asked to test the hypothesis, $H_o$, that the two stock have equal variances and that any observed difference is due to chance (i.e. $\\sigma_1^2 = \\sigma_2^2$). The alternate hypothesis, $H_a$, states that stock 1 has greater variability than stock 2 (i.e. $\\sigma_1^2 > \\sigma_2^2$).\n\nSince we are given summary statistics of the samples and not the full dataset, we cannot use the `var.test()` function which requires the full dataset as input. Instead, we will compute the $F$ ratio and observed probabilities separately.\n\nThe $F$ ratio is:\n\n$$\nF = \\frac{(.76)^2}{(.46)^2} = 2.73\n$$\n\n\nThe degrees of freedom are $(25 - 1) = 24$ for both samples.\n\nThe probability of getting a test statistic as extreme as ours can be computed using the `pf()` function:\n\n\n::: {.cell}\n\n```{.r .cell-code}\npf( 2.73, 24, 24, lower.tail = FALSE)\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n[1] 0.008502252\n```\n:::\n:::\n\n\nNote that we are using the  `lower.tail = FALSE` option since our alternate hypothesis is that $\\sigma_1^2 > \\sigma_2^2$. This gives us a probability of $0.008$, in other words, if the difference between stock 1 and stock 2 were explained by chance variability alone, there would be lest than a 1% chance of computing a $F$ ratio as extreme as ours. We can safely reject $H_o$ and state that the observed difference is real and that stock 1 has greater daily variability than stock 2.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![](F_test_files/figure-html/unnamed-chunk-7-1.png){width=672}\n:::\n:::\n\n\n\n# References\n\nFreedman D.A., Robert Pisani, Roger Purves. _Statistics_, 4th edition, 2007.  \nMcClave J.T., Dietrich F.H., _Statistics_, 4th edition, 1988.  \n\n-----\n\n**Session Info**:\n\n\n::: {.cell-output-display}\n**R version 4.2.1 (2022-06-23 ucrt)**\n\n**Platform:** x86_64-w64-mingw32/x64 (64-bit) \n\n\n**attached base packages:** \n_stats_, _graphics_, _grDevices_, _utils_, _datasets_, _methods_ and _base_\n\n**loaded via a namespace (and not attached):** \n_Rcpp(v.1.0.9)_, _digest(v.0.6.29)_, _Rttf2pt1(v.1.3.8)_, _jsonlite(v.1.8.0)_, _magrittr(v.2.0.3)_, _evaluate(v.0.16)_, _rlang(v.1.0.4)_, _stringi(v.1.7.8)_, _cli(v.3.3.0)_, _rstudioapi(v.0.14)_, _extrafontdb(v.1.0)_, _rmarkdown(v.2.16)_, _extrafont(v.0.18)_, _tools(v.4.2.1)_, _pander(v.0.6.5)_, _stringr(v.1.4.1)_, _xfun(v.0.32)_, _yaml(v.2.3.5)_, _fastmap(v.1.1.0)_, _compiler(v.4.2.1)_, _htmltools(v.0.5.3)_ and _knitr(v.1.40)_\n:::\n\n\n\n[1]: http://www.amstat.org/sections/srms/pamphlet.pdf\n",
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