From 7900c2ccd28e885104b97e6acc9902834e8c2b1f Mon Sep 17 00:00:00 2001 From: ethanweed Date: Fri, 3 May 2024 14:56:04 +0200 Subject: [PATCH] Update documentation --- 05.05-anova2.html | 178 ++++++++++++++++++++-- _sources/05.05-anova2.ipynb | 287 +++++++++++++++++++++++++++++++++--- searchindex.js | 2 +- 3 files changed, 434 insertions(+), 33 deletions(-) diff --git a/05.05-anova2.html b/05.05-anova2.html index 7ccd69d4..ea3863ce 100644 --- a/05.05-anova2.html +++ b/05.05-anova2.html @@ -2309,8 +2309,8 @@

17.5.2. ANOVA with binary factors as a r # Here we define our model. -model = ols('grade ~ attend + reading', data=rtfm2).fit() -sm.stats.anova_lm(model, typ=2).round(4) +model = ols('grade ~ attend + reading', data=rtfm1).fit() +sm.stats.anova_lm(model, typ=2).round(3) @@ -2344,15 +2344,15 @@

17.5.2. ANOVA with binary factors as a r attend 648.0 1.0 - 21.6000 - 0.0056 + 21.600 + 0.006 reading 1568.0 1.0 - 52.2667 - 0.0008 + 52.267 + 0.001 Residual @@ -2422,6 +2422,12 @@

17.5.2. ANOVA with binary factors as a r +

As you can see, the intercept term \(b_0\) acts like a kind of “baseline” grade that you would expect from those students who don’t take the time to attend class or read the textbook. Similarly, \(b_1\) represents the boost that you’re expected to get if you come to class, and \(b_2\) represents the boost that comes from reading the textbook. In fact, if this were an ANOVA you might very well want to characterise \(b_1\) as the main effect of attendance, and \(b_2\) as the main effect of reading! In fact, for a simple \(2 \times 2\) ANOVA that’s exactly how it plays out.

+

Okay, now that we’re really starting to see why ANOVA and regression are basically the same thing, let’s actually run our regression using a regression function to convince ourselves that this is really true. And this brings us to the first reason for using statsmodels for this example rather than pingouin. If you look back at the code we used to run our ANOVA, you will see that it was this:

+

model = ols('grade ~ attend + reading', data=rtfm2).fit() +`sm.stats.anova_lm(model, typ=2).round(4)```

+

That is to say, we already ran a regression! Our variable model literally is a linear regression, and all we did with sm.stats.anova_lm was dress up the results in an ANOVA costume[7]

+

Let’s take a closer look at the output of that regression we did. statsmodels spits out a lot of information, but I just want to zoom in on the part that is relevant for us now, which can be found like this:

model.summary().tables[1]
@@ -2435,19 +2441,167 @@ 
17.5.2. ANOVA with binary factors as a r
- + - + - + - +
coef std err t P>|t| [0.025 0.975] coef std err t P>|t| [0.025 0.975]
Intercept 43.5000 3.354 12.969 0.000 34.878 52.122Intercept 43.5000 3.354 12.969 0.000 34.878 52.122
attend[T.yes] 18.0000 3.873 4.648 0.006 8.044 27.956attend 18.0000 3.873 4.648 0.006 8.044 27.956
reading[T.yes] 28.0000 3.873 7.230 0.001 18.044 37.956reading 28.0000 3.873 7.230 0.001 18.044 37.956
+

There’s a few interesting things to note here. First, notice that the intercept term is 43.5, which is close to the “group” mean of 42.5 observed for those two students who didn’t read the text or attend class. Second, notice that we have the regression coefficient of \(b_1 = 18.0\) for the attendance variable, suggesting that those students that attended class scored 18% higher than those who didn’t. So our expectation would be that those students who turned up to class but didn’t read the textbook would obtain a grade of \(b_0 + b_1\), which is equal to \(43.5 + 18.0 = 61.5\). Again, this is similar to the observed group mean of 62.5. You can verify for yourself that the same thing happens when we look at the students that read the textbook.

+

Actually, we can push a little further in establishing the equivalence of our ANOVA and our regression. Look at the \(p\)-values associated with the attend variable and the reading variable in the regression output. They’re identical to the ones we encountered earlier when running the ANOVA. This might seem a little surprising, since the test used when running our regression model calculates a \(t\)-statistic and the ANOVA calculates an \(F\)-statistic. However, if you can remember all the way back to the chapter on probability, I mentioned that there’s a relationship between the \(t\)-distribution and the \(F\)-distribution: if you have some quantity that is distributed according to a \(t\)-distribution with \(k\) degrees of freedom and you square it, then this new squared quantity follows an \(F\)-distribution whose degrees of freedom are 1 and \(k\). We can check this with respect to the \(t\) statistics in our regression model. For the attend variable we get a \(t\) value of 4.648. If we square this number we end up with 21.604, which is identical to the corresponding \(F\) statistic in our ANOVA. I love it when a plan comes together.

+

I mentioned there was a second reason I didn’t use pingouin for this example. This is because as far as I can tell, when performing an ANOVA pingouin always calculated not only the main effects, but also the interaction, thus giving slightly different results. In order to keep things simple (and maintain parity with LSR, I decided to go with statsmodels and not specify any interactions. Just to be sure though, let’s run the ANOVA with pingouin, and then run the regression in statsmodels with a little ANOVA dressing on top, and confirm that we get the same result:

+
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pg.anova(dv='grade', between=['attend', 'reading'], data=rtfm1).round(3)
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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SourceSSDFMSFp-uncnp2
0attend648.01648.018.2540.0130.820
1reading1568.011568.044.1690.0030.917
2attend * reading8.018.00.2250.6600.053
3Residual142.0435.5NaNNaNNaN
+
+
+
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+
model = ols('grade ~ attend + reading + attend:reading', data=rtfm1).fit()
+sm.stats.anova_lm(model, typ=2).round(3)
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+
+
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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
sum_sqdfFPR(>F)
attend648.01.018.2540.013
reading1568.01.044.1690.003
attend:reading8.01.00.2250.660
Residual142.04.0NaNNaN
+
+
+

Yup. Same thing.

17.5.3. Changing the baseline category#

@@ -2528,6 +2682,10 @@

17.9. Summary[6]

There could be all sorts of reasons for doing this, I would imagine.

+

diff --git a/_sources/05.05-anova2.ipynb b/_sources/05.05-anova2.ipynb index 7c175e7b..7642eb31 100644 --- a/_sources/05.05-anova2.ipynb +++ b/_sources/05.05-anova2.ipynb @@ -2726,7 +2726,7 @@ }, { "cell_type": "code", - "execution_count": 68, + "execution_count": 84, "id": "d8afc8e5-3a97-4a88-9090-75c814abd4b3", "metadata": {}, "outputs": [ @@ -2762,15 +2762,15 @@ " attend\n", " 648.0\n", " 1.0\n", - " 21.6000\n", - " 0.0056\n", + " 21.600\n", + " 0.006\n", " \n", " \n", " reading\n", " 1568.0\n", " 1.0\n", - " 52.2667\n", - " 0.0008\n", + " 52.267\n", + " 0.001\n", " \n", " \n", " Residual\n", @@ -2784,13 +2784,13 @@ "
" ], "text/plain": [ - " sum_sq df F PR(>F)\n", - "attend 648.0 1.0 21.6000 0.0056\n", - "reading 1568.0 1.0 52.2667 0.0008\n", - "Residual 150.0 5.0 NaN NaN" + " sum_sq df F PR(>F)\n", + "attend 648.0 1.0 21.600 0.006\n", + "reading 1568.0 1.0 52.267 0.001\n", + "Residual 150.0 5.0 NaN NaN" ] }, - "execution_count": 68, + "execution_count": 84, "metadata": {}, "output_type": "execute_result" } @@ -2801,8 +2801,8 @@ "\n", "# Here we define our model.\n", "\n", - "model = ols('grade ~ attend + reading', data=rtfm2).fit()\n", - "sm.stats.anova_lm(model, typ=2).round(4)" + "model = ols('grade ~ attend + reading', data=rtfm1).fit()\n", + "sm.stats.anova_lm(model, typ=2).round(3)" ] }, { @@ -2897,29 +2897,62 @@ "| | **yes** | $b_0 + b_1$\" | $b_0 + b_1 + b_2$ |" ] }, + { + "cell_type": "markdown", + "id": "104edecf-15b1-4e38-a348-660df6fb80b4", + "metadata": {}, + "source": [ + "As you can see, the intercept term $b_0$ acts like a kind of \"baseline\" grade that you would expect from those students who don't take the time to attend class or read the textbook. Similarly, $b_1$ represents the boost that you're expected to get if you come to class, and $b_2$ represents the boost that comes from reading the textbook. In fact, if this were an ANOVA you might very well want to characterise $b_1$ as the main effect of attendance, and $b_2$ as the main effect of reading! In fact, for a simple $2 \\times 2$ ANOVA that's *exactly* how it plays out. " + ] + }, + { + "cell_type": "markdown", + "id": "5fa1588c-c18e-4cbf-9213-c275a47f0ff7", + "metadata": {}, + "source": [ + "Okay, now that we're really starting to see why ANOVA and regression are basically the same thing, let's actually run our regression using a regression function to convince ourselves that this is really true. And this brings us to the first reason for using `statsmodels` for this example rather than `pingouin`. If you look back at the code we used to run our ANOVA, you will see that it was this:\n", + "\n", + "`model = ols('grade ~ attend + reading', data=rtfm2).fit()`\n", + "`sm.stats.anova_lm(model, typ=2).round(4)```\n", + "\n", + "That is to say, we already ran a regression! Our variable `model` literally is a linear regression, and all we did with `sm.stats.anova_lm` was dress up the results in an ANOVA costume[^halloween]\n", + "\n", + "Let's take a closer look at the output of that regression we did. `statsmodels` spits out a lot of information, but I just want to zoom in on the part that is relevant for us now, which can be found like this:\n", + "\n", + "[^halloween]: Wait a minute... I just got an idea for my next Halloween costume 🎃" + ] + }, { "cell_type": "code", - "execution_count": 74, + "execution_count": 83, "id": "1cc7a2ed-ee68-40f9-b16f-b25478a50964", "metadata": { "scrolled": true }, "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/ethan/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/scipy/stats/_stats_py.py:1736: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8\n", + " warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n" + ] + }, { "data": { "text/html": [ "\n", "\n", - " \n", + " \n", "\n", "\n", - " \n", + " \n", "\n", "\n", - " \n", + " \n", "\n", "\n", - " \n", + " \n", "\n", "
coef std err t P>|t| [0.025 0.975] coef std err t P>|t| [0.025 0.975]
Intercept 43.5000 3.354 12.969 0.000 34.878 52.122Intercept 43.5000 3.354 12.969 0.000 34.878 52.122
attend[T.yes] 18.0000 3.873 4.648 0.006 8.044 27.956attend 18.0000 3.873 4.648 0.006 8.044 27.956
reading[T.yes] 28.0000 3.873 7.230 0.001 18.044 37.956reading 28.0000 3.873 7.230 0.001 18.044 37.956
" ], @@ -2927,11 +2960,11 @@ "\\begin{center}\n", "\\begin{tabular}{lcccccc}\n", "\\toprule\n", - " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", - "\\textbf{Intercept} & 43.5000 & 3.354 & 12.969 & 0.000 & 34.878 & 52.122 \\\\\n", - "\\textbf{attend[T.yes]} & 18.0000 & 3.873 & 4.648 & 0.006 & 8.044 & 27.956 \\\\\n", - "\\textbf{reading[T.yes]} & 28.0000 & 3.873 & 7.230 & 0.001 & 18.044 & 37.956 \\\\\n", + "\\textbf{Intercept} & 43.5000 & 3.354 & 12.969 & 0.000 & 34.878 & 52.122 \\\\\n", + "\\textbf{attend} & 18.0000 & 3.873 & 4.648 & 0.006 & 8.044 & 27.956 \\\\\n", + "\\textbf{reading} & 28.0000 & 3.873 & 7.230 & 0.001 & 18.044 & 37.956 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\end{center}" @@ -2940,7 +2973,7 @@ "" ] }, - "execution_count": 74, + "execution_count": 83, "metadata": {}, "output_type": "execute_result" } @@ -2949,6 +2982,216 @@ "model.summary().tables[1]" ] }, + { + "cell_type": "markdown", + "id": "998f52e6-6138-4b7a-a5d4-57410708225d", + "metadata": {}, + "source": [ + "There's a few interesting things to note here. First, notice that the intercept term is 43.5, which is close to the \"group\" mean of 42.5 observed for those two students who didn't read the text or attend class. Second, notice that we have the regression coefficient of $b_1 = 18.0$ for the attendance variable, suggesting that those students that attended class scored 18\\% higher than those who didn't. So our expectation would be that those students who turned up to class but didn't read the textbook would obtain a grade of $b_0 + b_1$, which is equal to $43.5 + 18.0 = 61.5$. Again, this is similar to the observed group mean of 62.5. You can verify for yourself that the same thing happens when we look at the students that read the textbook.\n", + "\n", + "Actually, we can push a little further in establishing the equivalence of our ANOVA and our regression. Look at the $p$-values associated with the `attend` variable and the `reading` variable in the regression output. They're identical to the ones we encountered earlier when running the ANOVA. This might seem a little surprising, since the test used when running our regression model calculates a $t$-statistic and the ANOVA calculates an $F$-statistic. However, if you can remember all the way back to the chapter on [probability](probability), I mentioned that there's a relationship between the $t$-distribution and the $F$-distribution: if you have some quantity that is distributed according to a $t$-distribution with $k$ degrees of freedom and you square it, then this new squared quantity follows an $F$-distribution whose degrees of freedom are 1 and $k$. We can check this with respect to the $t$ statistics in our regression model. For the `attend` variable we get a $t$ value of 4.648. If we square this number we end up with 21.604, which is identical to the corresponding $F$ statistic in our ANOVA. [I love it when a plan comes together.](https://www.youtube.com/watch?v=NsUFBm1uENs)\n" + ] + }, + { + "cell_type": "markdown", + "id": "76fcaa5a-c7b3-497d-9fe3-5a01701e199f", + "metadata": {}, + "source": [ + "I mentioned there was a second reason I didn't use `pingouin` for this example. This is because as far as I can tell, when performing an ANOVA `pingouin` always calculated not only the main effects, but also the interaction, thus giving slightly different results. In order to keep things simple (and maintain parity with [LSR](http://learningstatisticswithr.com/), I decided to go with `statsmodels` and not specify any interactions. Just to be sure though, let's run the ANOVA with `pingouin`, and then run the regression in `statsmodels` with a little ANOVA dressing on top, and confirm that we get the same result:" + ] + }, + { + "cell_type": "code", + "execution_count": 99, + "id": "ec480505-abcd-43fa-abdd-a5622915820a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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3Residual142.0435.5NaNNaNNaN
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" + ], + "text/plain": [ + " Source SS DF MS F p-unc np2\n", + "0 attend 648.0 1 648.0 18.254 0.013 0.820\n", + "1 reading 1568.0 1 1568.0 44.169 0.003 0.917\n", + "2 attend * reading 8.0 1 8.0 0.225 0.660 0.053\n", + "3 Residual 142.0 4 35.5 NaN NaN NaN" + ] + }, + "execution_count": 99, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pg.anova(dv='grade', between=['attend', 'reading'], data=rtfm1).round(3)" + ] + }, + { + "cell_type": "code", + "execution_count": 98, + "id": "b8cbf2e1-be49-4206-a2ea-35dc01501053", + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " sum_sq df F PR(>F)\n", + "attend 648.0 1.0 18.254 0.013\n", + "reading 1568.0 1.0 44.169 0.003\n", + "attend:reading 8.0 1.0 0.225 0.660\n", + "Residual 142.0 4.0 NaN NaN" + ] + }, + "execution_count": 98, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "model = ols('grade ~ attend + reading + attend:reading', data=rtfm1).fit()\n", + "sm.stats.anova_lm(model, typ=2).round(3)" + ] + }, + { + "cell_type": "markdown", + "id": "d8e11264-5512-4524-9ae5-f91f6fd8497d", + "metadata": {}, + "source": [ + "Yup. Same thing." + ] + }, { "attachments": {}, "cell_type": "markdown", diff --git a/searchindex.js b/searchindex.js index 94c41c90..2e70b950 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["01.01-intro", "01.02-studydesign", "02.01-getting_started_with_python", "02.02-more_python_concepts", "03.01-descriptives", "03.02-drawing_graphs", "03.03-pragmatic_matters", "03.04-basic_programming", "04.01-intro-to-probability", "04.02-probability", "04.03-estimation", "04.04-hypothesis-testing", "05.01-chisquare", "05.02-ttest", "05.03-anova", "05.04-regression", "05.05-anova2", "06.01-bayes", "06.02-epilogue", "bibliography", "landingpage"], "filenames": ["01.01-intro.ipynb", "01.02-studydesign.ipynb", "02.01-getting_started_with_python.ipynb", "02.02-more_python_concepts.ipynb", "03.01-descriptives.ipynb", "03.02-drawing_graphs.ipynb", "03.03-pragmatic_matters.ipynb", "03.04-basic_programming.ipynb", "04.01-intro-to-probability.ipynb", "04.02-probability.ipynb", "04.03-estimation.ipynb", "04.04-hypothesis-testing.ipynb", "05.01-chisquare.ipynb", "05.02-ttest.ipynb", "05.03-anova.ipynb", "05.04-regression.ipynb", "05.05-anova2.ipynb", "06.01-bayes.ipynb", "06.02-epilogue.ipynb", "bibliography.md", "landingpage.md"], "titles": ["1. 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