From 5d9debb7d63148abef1cba489362d675557406bd Mon Sep 17 00:00:00 2001 From: Richard Evans Date: Fri, 10 Nov 2023 00:47:47 -0700 Subject: [PATCH] Updated LogisticReg.md --- data/basic_empirics/logit/titanic_clean.csv | 713 ++++++++++++++++++ docs/book/basic_empirics/LogisticReg.md | 223 +++++- images/basic_empirics/logit/age_boxplot.png | Bin 0 -> 258285 bytes .../basic_empirics/logit/survived_count.png | Bin 0 -> 106239 bytes 4 files changed, 935 insertions(+), 1 deletion(-) create mode 100644 data/basic_empirics/logit/titanic_clean.csv create mode 100644 images/basic_empirics/logit/age_boxplot.png create mode 100644 images/basic_empirics/logit/survived_count.png diff --git a/data/basic_empirics/logit/titanic_clean.csv b/data/basic_empirics/logit/titanic_clean.csv new file mode 100644 index 0000000..9eb3044 --- /dev/null +++ b/data/basic_empirics/logit/titanic_clean.csv @@ -0,0 +1,713 @@ +,Survived,SibSp,Parch,Fare,Age,male,Q,S +0,0,1,0,7.25,22.0,1,0,1 +1,1,1,0,71.2833,38.0,0,0,0 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+886,0,0,0,13.0,27.0,1,0,1 +887,1,0,0,30.0,19.0,0,0,1 +889,1,0,0,30.0,26.0,1,0,0 +890,0,0,0,7.75,32.0,1,1,0 diff --git a/docs/book/basic_empirics/LogisticReg.md b/docs/book/basic_empirics/LogisticReg.md index a4837b5..2e352f4 100644 --- a/docs/book/basic_empirics/LogisticReg.md +++ b/docs/book/basic_empirics/LogisticReg.md @@ -205,7 +205,7 @@ The variable descriptions are the following: * `Cabin`: Cabin * `Embarked`: Port of embarkation (C = Cherbourg; Q = Queenstown; S = Southampton) -Let's first check that our target variable, `Survived`, is binary. Since we are building a model to predict survival of passangers from the Titanic, our target is going to be the `Survived` variable from the titanic dataframe. To make sure that it is a binary variable, let's use Seaborn's `countplot()` function. +Let's first check that our target variable, `Survived`, is binary. Since we are building a model to predict survival of passangers from the Titanic, our target is going to be the `Survived` variable from the titanic dataframe. To make sure that it is a binary variable, let's use Panda's `.value_counts()` method and plot the counts in a `matplotlib.pyplot` bar chart. ```{code-cell} ipython3 :tags: [] @@ -213,6 +213,227 @@ Let's first check that our target variable, `Survived`, is binary. Since we are titanic['Survived'].value_counts() ``` +```{code-cell} ipython3 +:tags: ["remove-output"] + +categories = titanic["Survived"].value_counts().index +counts = titanic["Survived"].value_counts().values +fig, ax = plt.subplots() +vbars = plt.bar( + ["0: Did not survive", "1: Survived"], counts, color=["blue", "green"], + alpha=0.8 +) +ax.yaxis.grid(color='gray', linestyle=':', linewidth=1, alpha=0.5) +plt.bar_label(vbars, label_type="edge") +plt.title(r"Counts for 0 and 1 for Survived variable") +plt.xlabel(r'Survived') +plt.ylabel(r'Count') +plt.show() +``` + +```{figure} ../../../images/basic_empirics/logit/survived_count.png +:height: 500px +:name: FigLogit_survived_count + +Counts for 0 and 1 for Survived variable +``` + +We can check for missing values in the DataFrame. It is easy to check for missing values by calling the `isnull()` method, and the `sum()` method off of the DataFrame to return a tally of all the `True` values that are returned by the `isnull()` method. + +```{code-cell} ipython3 +:tags: [] + +titanic.isnull().sum() +``` +How many observations are there in the DataFrame? + +```{code-cell} ipython3 +:tags: [] + +titanic.info() +``` + +**Model selection and missing values** + +The variable `Cabin` is missing data in the majority of its observations, so we should not include that in our analysis (although that would be an interesting variable to have). We can also probably exclude `Name`, `Ticket` (ticket number). But we will include all the other variables. + +* `Survived`: This variable is obviously relevant. +* `Pclass`: Does a passenger's class on the boat affect their survivability? +* `Sex`: Could a passenger's gender impact their survival rate? +* `Age`: Does a person's age impact their survival rate? +* `SibSp`: Does the number of relatives on the boat (that are siblings or a spouse) affect a person survivability? Probably. +* `Parch`: Does the number of relatives on the boat (that are children or parents) affect a person survivability? Probably. +* `Fare`: Does the fare a person paid effect his survivability? Maybe. +* `Embarked`: Does a person's point of embarkation matter? It depends on how the boat was filled. + +```{code-cell} ipython3 +:tags: [] + +titanic_data = titanic.drop(['PassengerId','Name','Ticket','Cabin'], axis=1) +titanic_data.head() +``` + +Now we have the dataframe reduced down to only relevant variables, but now we need to deal with the missing values in the age variable. + +**Imputing missing values** + +Let's look at how passenger age is related to their class as a passenger on the boat. + +```{code-cell} ipython3 +:tags: [] + +titanic_data['Pclass'].value_counts() +``` + +```{code-cell} ipython3 +:tags: ["remove-output"] + +# Create a list of three NumPy arrays of Age data by Pclass +age_by_pclass_dtalst = [] +for pclass_val in range(1, 4): + age_by_pclass_vec = titanic_data["Age"][ + titanic_data["Pclass"]==pclass_val + ].values + # Have to remove nan values for the boxplot to work + age_by_pclass_vec_nonan = age_by_pclass_vec[ + ~np.isnan(age_by_pclass_vec) + ] + age_by_pclass_dtalst.append(age_by_pclass_vec_nonan) + +labels = ["1: First class", "2: Second class", "3: Third class"] + +fig, ax = plt.subplots() +boxplot1 = plt.boxplot( + age_by_pclass_dtalst, + notch=True, # Notch shape + vert=True, # Vertical box alignment + patch_artist=True, # Fill boxes with color + showmeans=True, # Show the mean value as a scatter point in each box + labels=labels +) +ax.yaxis.grid(color='gray', linestyle=':', linewidth=1, alpha=0.5) +plt.title(r"Box and whisker plots for age by passenger class") +plt.xlabel(r'Passenger class ($Pclass$)') +plt.ylabel(r'Age') + +# fill boxes with colors +colors = ['pink', 'lightblue', 'lightgreen'] +for patch, color in zip(boxplot1['boxes'], colors): + patch.set_facecolor(color) + +plt.show() +``` + +```{figure} ../../../images/basic_empirics/logit/age_boxplot.png +:height: 500px +:name: FigLogit_age_boxplot + +Box and whisker plots for age by passenger class +``` + +The box and whisker plots in {numref}`Figure %s ` are a nice way to visualize where the data live. The two ends of the colored box show the first quartile (25th percentil) to the third quartile (75th percentile), with the notched line representing the median (50th percentile). A scatter point great triangle shows the mean. The whiskers extend to 1.5x the inter-quartile range (IQR) of ages, with outliers beyond the whiskers shown as scatter points. An alternative would be to show a histogram. + +Roughly speaking, we could say that the younger a passenger is, the more likely they are to be in 3rd class. The older a passenger is, the more likely they are to be in 1st class. So there is a loose relationship between these variables. So let's write a function that approximates a passengers age, based on their class. From the box plot, it looks like the average age of 1st class passengers is about 37, 2nd class passengers is 29, and 3rd class pasengers is 24. + +So let's write a function that finds each null value in the `Age` variable, and for each null, checks the value of the Pclass and assigns an age value according to the average age of passengers in that class. + +```{code-cell} ipython3 +:tags: [] + +median_ages = titanic_data.groupby("Pclass")["Age"].median().values + +# Create new variable Age_imputed and impute the median ages for missing values +titanic_data["Age_imputed"] = titanic_data["Age"] +for pclass_val in range(1, 4): + row_indexer = ( + titanic_data["Age"].isnull() * titanic_data["Pclass"] == pclass_val + ) + titanic_data.loc[row_indexer, "Age_imputed"] = median_ages[pclass_val - 1] + imputed_obs = row_indexer.sum() + print("Imputed", imputed_obs, "Age values with Pclass=", pclass_val) + +titanic_data.isnull().sum() +``` + +There are 2 null values in the `Embarked` variable. We can drop those 2 records without loosing too much important information from our dataset, so we will do that. + +```{code-cell} ipython3 +:tags: [] + +titanic_data.dropna(inplace=True) +titanic_data.isnull().sum() +``` + +```{code-cell} ipython3 +:tags: [] + +titanic_data.head() +``` + +**Converting categorical variables to a dummy indicators** + +The next thing we need to do is reformat our variables so that they work with the model. Specifically, we need to reformat the `Sex` and `Embarked` variables into numeric categorical variables. + + +(Sec_LogLogitInterpret)= +#### Interpreting coefficients (log odds ratio) + +The odds ratio in the logistic model is provides a nice way to interpret logit model coefficients. Let $z\equiv X^T\beta = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}$. The logistic model is stated by the probability that the binary categorical dependent variable equals one $y_i=1$. +\begin{equation} + P(y_i=1|X,\theta) = \frac{e^z}{1 + e^z} +\end{equation} +Given this equation, we know that the probability of the dependent variable being zero $y_i=0$ is just one minus the probability above. +\begin{equation} + P(y_i=0|X,\theta) = 1 - P(y_i=1|X,\theta) = 1 - \frac{e^z}{1 + e^z} = \frac{1}{1 + e^z} +\end{equation} + +The odds ratio is a common way of expressing the probability of an event versus all other events. For example, if the probability of your favorite team winning a game is $P(win)=0.8$, then we know that the probability of your favorite team losing that game is $P(lose)=1-P(win)=0.2$. The odds ratio is the ratio of these two probabilities. +\begin{equation} + \frac{P(win)}{P(lose)} = \frac{P(win)}{1 - P(win)} = \frac{0.8}{0.2} = \frac{4}{1} \quad\text{or}\quad 4 +\end{equation} +The odds ratio tells you that the probability of your team winning is four times as likely as your team losing. A gambler would say that your odds are 4-to-1. Another way of saying it is that your team will win four out of five times and will lose 1 out of five times. + +In the logistic model, the odds ratio reduces the problem nicely. +\begin{equation} + \frac{P(y_i=1|X,\theta)}{1 - P(y_i=1|X,\theta)} = \frac{\frac{e^z}{1 + e^z}}{\frac{1}{1 + e^z}} = e^z +\end{equation} +If we take the log of both sides, we see that the log odds ratio is equal to the linear predictor $z\equiv X^T\beta = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}$. +\begin{equation} + \ln\left(\frac{P(y_i=1|X,\theta)}{1 - P(y_i=1|X,\theta)}\right) = z = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i} +\end{equation} + +So the interpretation of the coeficients $\beta_k$ is that a one-unit increase of the variable $x_{k,i}$ increases the odds ratio or the odds of $y_i=1$ by $\beta_{k,i}$ percent. + + +(Sec_LogMultiNomLogit)= +## Multinomial Logit +The multinomial logit model is a natural extension of the logit model. In contrast to the logit model in which the dependent variable has only two categories, the multinomial logit model accomodates $J\geq2$ categories in the dependent variable. Let $\eta_j$ be the linear predictor for the $j$th category. +$$ \eta_j\equiv \beta_{j,0} + \beta_{j,1}x_{1,i} + ...\beta_{j,K}x_{K,i} \quad\forall y_i = j $$ + +The multinomial logit model gives the probability of $y_i=j$ relative to some reference category $J$ that is left out. +\begin{equation} + Pr(y_i=j|X,\theta) = \frac{e^{\eta_j}}{1 + \sum_v^{J-1}e^{\eta_v}} \quad\text{for}\quad 1\leq j\leq J-1 +\end{equation} + +Once the $J-1$ sets of coefficients are estimated, the final $J$th set of coefficients are a residual based on the following expression. +\begin{equation} + Pr(y_i=J|X,\theta) = \frac{1}{1 + \sum_v^{J-1}e^{\eta_v}} +\end{equation} + +The analogous log odds ratio interpretation applies to the multinomial logit model. +\begin{equation} + \ln\left(\frac{Pr(y_i=j|X,\theta)}{Pr(y_i=J|X,\theta)}\right) = \eta_j = \beta_{j,0} + \beta_{j,1}x_{1,i} + ...\beta_{j,K}x_{K,i} \quad\text{for}\quad 1\leq j \leq J-1 +\end{equation} +This is the odds ratio of $y_i=j$ relative to $y_i=J$. The interpretation of the $\beta_{j,k}$ coefficient is the predicted percentage change in the log odds ratio of $y_i=j$ to $y_i=J$ from a one-unit increase in variable $x_{k,i}$. + +TODO: list code for Iris example + + +(Sec_LogExercises)= +## Exercises + +Put exercises here. + (SecLogFootnotes)= ## Footnotes diff --git a/images/basic_empirics/logit/age_boxplot.png b/images/basic_empirics/logit/age_boxplot.png new file mode 100644 index 0000000000000000000000000000000000000000..798d5a2401077ac686959cfe5d1d7b265403f892 GIT binary patch literal 258285 zcma%j2|UzW`*^#Hq9{w2QMMvudyyD!B-xUCp8S;msBFd?a%ZG`N^ z7;BbvnTE*Pl(7thvHy>{@A7-!dq1DQ&u7eczTb1s_c_mb_VYXkd*iz9F}Ble`}XZS zrg!bi&3*e0`|jI!=PiZ|DP z(oxXNVH3}#p`to4ZjhWT)ZmV;bg&;skKa{#{rsUobG9Qc``b9CxG?my9m3O?cjD;> z$!Vh#PI13LwJyB5yiYci2fM4teY^Le>aUZ(WQ8a=p6|fiI2kACI_}5zvU~ExYR6Sb zsN(DxzkdaHhvO>=+xNZ|vYa&h?FdpSgb zJ@v?~8v8n1zN-e#EchIMChHX^`(X+Ra}pr?k&~sKvyst0DRBMEzC-&@?>h{x_Jfbc zet~~oU)_Ih-@zaE5A53)<+|_CpKY##@4deS@Y(D0^ZQ^@%+d! 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