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+Exercise 12: Question 1¶
Dataset- chichwts.txt¶
This dataset describes the attributes of chicks. The first column contains the weights of the chicks in a numeric value and the second column contains a character string which describes the type of feed they received.
+Import Necessary Packages and Data¶
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+In [4]:
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+#Import packages
+import numpy
+import pandas
+import scipy
+import scipy.integrate as spint
+from scipy.stats import norm
+from scipy.optimize import minimize
+from scipy.stats import chi2
+import plotnine
+from plotnine import *
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+#Load Data
+chicken=pandas.read_csv("chickwts.txt", sep=",")
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+Part 1: Generate a Plot to Summarize Data¶
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+In [5]:
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+#Generate plot that shows the weights of chicks vs. feed type(scatter)
+plot1= ggplot(chicken,aes(x="feed",y="weight"))+geom_dotplot(binaxis="y",stackdir="center", stackratio=0.5, dotsize=0.2)+theme_classic()
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+print plot1
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+#Generate plot that shows average weight vs. feed type (bar)
+plot2= ggplot(chicken, aes(y="weight",x="feed"))+geom_bar(stat="summary",fun_y=numpy.mean)
+print plot2
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+Part 2: Null Hypotheses for Difference in Chick Weight (Soybean vs. Sunflower)¶
Hypotheses¶
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- Null Hypothesis: There is no difference in chick weight when fed soybean or sunflower seed. +
- Alternative Hypothesis: There is a significant difference in chick weight between those fed soybean feed and those fed sunflower seed. +
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+In [6]:
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+#Subset the Data to Only Have the Types of Feed we are Interested in
+chicksub=chicken.loc[chicken.feed.isin(['soybean', 'sunflower']),:]
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+#Make Dataframe for Modeling
+chickFrame= pandas.DataFrame({'y':chicksub.weight,'x':0})
+chickFrame.loc[chicksub.feed=='sunflower','x']=1
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+Part 3: Test Null Hypothesis Using Likelihood Ratio Test¶
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- Step 1: Define Null Model +
- Step 2: Define Alternative Model +
- Step 3: Run/Generate Likelihood Ratio Tests +
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+In [12]:
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+#Define Null
+def nllikeNull(pNull,obsNull):
+ B0Null=pNull[0]
+ sigmaNull=pNull[1]
+ expectedNull=B0Null
+ nllNull=-1*norm(expectedNull,sigmaNull).logpdf(obsNull.y).sum()
+ return nllNull
+
+#Define Alternative
+def nlllikeAlt(pAlt,obsAlt):
+ B0Alt=pAlt[0]
+ B1Alt=pAlt[1]
+ sigmaAlt=pAlt[2]
+ expectedAlt=B0Alt+B1Alt*obsAlt.x
+ nllAlt=-1*norm(expectedAlt,sigmaAlt).logpdf(obsAlt.y).sum()
+ return nllAlt
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+In [8]:
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+#Null Model
+initialGuessNull=numpy.array([1,1])
+fitNullChickFrame=minimize(nllikeNull,initialGuessNull, method="Nelder-Mead",options={'disp':True},args=chickFrame)
+print("Estimated Parameters: Null Model")
+print(fitNullChickFrame.x)
+print("NLL: Null Model")
+nllNullChickFrame=fitNullChickFrame.fun
+#Print NLL value for Null Model
+print(nllNullChickFrame)
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+In [13]:
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+#Alternative Model
+initialGuessAlt=numpy.array([1,1,1])
+fitAltChickFrame=minimize(nlllikeAlt,initialGuessAlt,method="Nelder-Mead",options={'disp':True},args=chickFrame)
+print("Estimated Parameters: Alternative Model")
+print(fitAltChickFrame.x)
+print("NLL: Alternative Model")
+nllAltChickFrame=fitAltChickFrame.fun
+#Print NLL value for Alternative Model
+print(nllAltChickFrame)
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+Part 4: Interpret Results of Likelihood Ratio Test¶
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- Step 1: Calculate D value +
- Step 2: Calculate p value +
Conclusions:¶
As our p-value was less than 0.05, we can report that our alternative hypothesis was correct. This means that there is a significant difference between the chicks who were fed soybean feed vs. those who were fed sunflower feed.
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+In [14]:
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+#Calculate D value
+DchickFrame=2*(nllNullChickFrame-nllAltChickFrame)
+print("D Value")
+print(DchickFrame)
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+#Calculate p value
+pChickFrame=1-scipy.stats.chi2.cdf(x=DchickFrame,df=1)
+print("p-value")
+print(pChickFrame)
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