AP Statistics Curriculum 2007 Contingency Fit - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T20:24:10Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Assumptions */

2014-10-27T11:47:15Z

‎Assumptions

Dinov: /* Assumptions */

2014-10-26T16:41:08Z

‎Assumptions

Dinov: /* ApoE and Alzheimer's disease (AD) */

2014-10-26T16:37:01Z

‎ApoE and Alzheimer's disease (AD)

Dinov at 16:32, 26 October 2014

2014-10-26T16:32:17Z

IvoDinov: /* ApoE and Alzheimer's disease (AD) */

2013-04-04T19:07:37Z

‎ApoE and Alzheimer's disease (AD)

IvoDinov: /* ApoE and Alzheimer's disease (AD) */

2013-04-04T19:06:10Z

‎ApoE and Alzheimer's disease (AD)

IvoDinov: added AD ApoE data

2013-04-04T19:02:58Z

added AD ApoE data

IvoDinov at 23:07, 30 January 2013

2013-01-30T23:07:49Z

IvoDinov: /* =Iris sepal and petal length */

2011-05-25T18:32:36Z

‎=Iris sepal and petal length

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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 where $O_i$ is the observed frequency and $E_i$ is the expected frequency for bin $1\leq i\leq k$. The expected counts may often be calculated by
-: $E_i = N\times(F(U_i) - F(L_i))$,
+: $E_i = k\times(F(U_i) - F(L_i))$,
-where ''N'' is the total sample size, ''F'' is the cumulative distribution function (CDF) for the distribution being tested, $U_i$ is the upper limit and $L_i$ is the lower limit for class ''i''.
+where ''k'' is the total sample size, ''F'' is the cumulative distribution function (CDF) for the distribution being tested, $U_i$ is the upper limit and $L_i$ is the lower limit for class ''i''.
 The chi-square test is sensitive to the choice of bins and the optimal choice for the bin-width may depend on the choice of the distribution. The chi-square test is valid if the data represent a random sample of at least 20 observations and the expected frequencies at each bin are at least 5. Otherwise, the distribution of the $\chi_o^2$ statistics is not guaranteed to be $\chi_{k - 1}^2$, in general. In particular, the test may not be valid for small samples. If the expected counts are less than five for some bins, you may need to combine bins together to increase these counts.

← Older revision		Revision as of 16:41, 26 October 2014
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	The chi-square test is sensitive to the choice of bins and the optimal choice for the bin-width may depend on the choice of the distribution. The chi-square test is valid if the data represent a random sample of at least 20 observations and the expected frequencies at each bin are at least 5. Otherwise, the distribution of the $\chi_o^2$ statistics is not guaranteed to be $\chi_{k - 1}^2$, in general. In particular, the test may not be valid for small samples. If the expected counts are less than five for some bins, you may need to combine bins together to increase these counts.		The chi-square test is sensitive to the choice of bins and the optimal choice for the bin-width may depend on the choice of the distribution. The chi-square test is valid if the data represent a random sample of at least 20 observations and the expected frequencies at each bin are at least 5. Otherwise, the distribution of the $\chi_o^2$ statistics is not guaranteed to be $\chi_{k - 1}^2$, in general. In particular, the test may not be valid for small samples. If the expected counts are less than five for some bins, you may need to combine bins together to increase these counts.
		+
		+	* [http://socr.ucla.edu/Applets.dir/SOCRCurveFitter.html Try the SOCR Polynomial curve modeling applet] to see how the chi-square test can be used to assess model quality.

	==Examples==		==Examples==

@@ Line 77: / Line 77: @@
 </center>
-Is there evidence of an association between genotype (alleles) and phenotype (disease)? What is the probability P(AD|ε2)?
+* Is there evidence of an association between genotype (alleles) and phenotype (disease)? You can use the [http://socr.umich.edu/html/ana/SOCR_Analyses.html SOCR Chi-Square Test for Association/Contingency Applet].
 ===Butterfly Hotspots===

@@ Line 21: / Line 21: @@
 ==Calculations==
-Suppose there were ''N = 1064'' data measurements with ''Observed(Tall) = 787'' and ''Observed(Dwarf) = 277''. These are the O’s (observed values). To calculate the E’s (expected values), we will take the hypothesized proportions under <math>H_o</math> and multiply them by the total sample size ''N''. Expected(Tall) = (0.75)(1064) = 798 and Expected(Dwarf) = (0.25)(1064) = 266. Quickly check to see if the expected total = N = 1064.
+Suppose there were ''N = 1064'' data measurements with ''Observed(Tall) = 787'' and ''Observed(Dwarf) = 277''. These are the O’s (observed values). To calculate the E’s (expected values), we will take the hypothesized proportions under $H_o$ and multiply them by the total sample size ''N''. Expected(Tall) = (0.75)(1064) = 798 and Expected(Dwarf) = (0.25)(1064) = 266. Quickly check to see if the expected total = N = 1064.
 * The Hypotheses:
-: <math>H_o</math>:P(tall) = 0.75 (No effect, follows a 3:1phenotypic ratio)
+: $H_o$:P(tall) = 0.75 (No effect, follows a 3:1phenotypic ratio)
 :: P(dwarf) = 0.25
-: <math>H_a</math>: P(tall)  ≠  0.75
+: $H_a$: P(tall)  ≠  0.75
 ::P(dwarf) ≠ 0.25
 * Test Statistics:
-:<math>\chi_o^2 = \sum_{all-categories}{(O-E)^2 \over E} \sim \chi_{(df=number\_of\_categories - 1)}^2</math>
+:$\chi_o^2 = \sum_{all-categories}{(O-E)^2 \over E} \sim \chi_{(df=number\_of\_categories - 1)}^2$
 * P-values and Critical values for the [http://socr.stat.ucla.edu/htmls/SOCR_Distributions.html Chi-Square Distribution may be easily computed using SOCR Distributions].
@@ Line 36: / Line 36: @@
 * Results:
 For the Mendel's Pea Experiment, we can compute the Chi-Square Test Statistics to be:
-: <math>\chi_o^2 = {(787-798)^2 \over 798}  + {(277-266)^2 \over 266} = 0.152+0.455=0.607</math>.
+: $\chi_o^2 = {(787-798)^2 \over 798}  + {(277-266)^2 \over 266} = 0.152+0.455=0.607$.
-: p-value=<math>P(\chi_{(df=1)}^2 > \chi_o^2)=0.436</math>
+: p-value=$P(\chi_{(df=1)}^2 > \chi_o^2)=0.436$
 * [[SOCR_EduMaterials_AnalysisActivities_Chi_Goodness |SOCR Chi-Square Calculations]]:
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 The chi-square goodness-of-fit test requires that the data is divided into ''k'' bins and the test statistic is defined as
-:<math>\chi_o^2 = \sum_{i=1}^k{(O_i-E_i)^2 \over E_i} \sim \chi_{(df=k - 1)}^2</math>,
+:$\chi_o^2 = \sum_{i=1}^k{(O_i-E_i)^2 \over E_i} \sim \chi_{(df=k - 1)}^2$,
-where <math>O_i</math> is the observed frequency and <math>E_i</math> is the expected frequency for bin <math>1\leq i\leq k</math>. The expected counts may often be calculated by
+where $O_i$ is the observed frequency and $E_i$ is the expected frequency for bin $1\leq i\leq k$. The expected counts may often be calculated by
-: <math>E_i = N\times(F(U_i) - F(L_i))</math>,
+: $E_i = N\times(F(U_i) - F(L_i))$,
-where ''N'' is the total sample size, ''F'' is the cumulative distribution function (CDF) for the distribution being tested, <math>U_i</math> is the upper limit and <math>L_i</math> is the lower limit for class ''i''.
+where ''N'' is the total sample size, ''F'' is the cumulative distribution function (CDF) for the distribution being tested, $U_i$ is the upper limit and $L_i$ is the lower limit for class ''i''.
-The chi-square test is sensitive to the choice of bins and the optimal choice for the bin-width may depend on the choice of the distribution. The chi-square test is valid if the data represent a random sample of at least 20 observations and the expected frequencies at each bin are at least 5. Otherwise, the distribution of the <math>\chi_o^2</math> statistics is not guaranteed to be <math>\chi_{k - 1}^2</math>, in general. In particular, the test may not be valid for small samples. If the expected counts are less than five for some bins, you may need to combine bins together to increase these counts.
+The chi-square test is sensitive to the choice of bins and the optimal choice for the bin-width may depend on the choice of the distribution. The chi-square test is valid if the data represent a random sample of at least 20 observations and the expected frequencies at each bin are at least 5. Otherwise, the distribution of the $\chi_o^2$ statistics is not guaranteed to be $\chi_{k - 1}^2$, in general. In particular, the test may not be valid for small samples. If the expected counts are less than five for some bins, you may need to combine bins together to increase these counts.
 ==Examples==
@@ Line 80: / Line 80: @@
 ===Butterfly Hotspots===
-A hotspot is defined as a <math>10 km^2</math> area that is species rich (heavily populated by the species of interest).  Suppose in a study of butterfly hotspots in a particular area of <math>10 km^2</math>, the number of butterfly hotspots in a sample of 2,588 is 165.  In theory, 5% of the areas should be butterfly hotspots.  Does the data provide evidence to suggest that the number of butterfly hotspots is increasing from the theoretical standards?  Test using <math>\alpha= 0.01</math>.
+A hotspot is defined as a $10 km^2$ area that is species rich (heavily populated by the species of interest).  Suppose in a study of butterfly hotspots in a particular area of $10 km^2$, the number of butterfly hotspots in a sample of 2,588 is 165.  In theory, 5% of the areas should be butterfly hotspots.  Does the data provide evidence to suggest that the number of butterfly hotspots is increasing from the theoretical standards?  Test using $\alpha= 0.01$.
 ===Cell-Phone Usage===

@@ Line 65: / Line 65: @@
 {| class="wikitable"
 |-
-| colspan="5"  align="center" | '''Estimated worldwide human allele frequencies of ApoE '''
+| colspan="6"  align="center" | '''Estimated worldwide human allele frequencies of ApoE '''
 |-
-| Allele || ε2 ||ε3 ||ε4
+| Allele || ε2 ||ε3 ||ε4||Total
 |-
-| General Frequency||8||78||14
+| General Frequency||8||78||14||100
 |-
-| AD Frequency||4||59||37
+| AD Frequency||4||59||37||100
 |}
 </center>

← Older revision		Revision as of 19:02, 4 April 2013
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	==Examples==		==Examples==
		+
		+	===ApoE and Alzheimer's disease (AD)===
		+
		+	ApoE (Apolipoprotein E) is a strong genetic risk factor for AD. About 40-65% of AD patients have at least one copy of the 4 allele, ApoE4, yet, at least 30% of patients with AD are ApoE4 negative and some ApoE4 homozygotes never develop the disease. People with two e4 alleles have up to 20 times the risk of developing AD. There is also evidence that the ApoE2 allele may serve a protective role in AD. Thus, the genotype most at risk for Alzheimer's disease and at an earlier age is the homozygous ApoE 4,4. The ApoE 3,4 genotype is at increased risk, genotype ApoE 3,3 is considered at normal risk for Alzheimer disease, and genotype ApoE 2,3 lowers the risk for Alzheimer disease.
		+
		+	Suppose we have a random sample of 100 AD patients and 100 asymptomatic age-matched controls. The table below illustrates the [http://en.wikipedia.org/wiki/Apolipoprotein_E#Alzheimer_disease expected distribution of the ApoE traits].
		+
		+	{\| class="wikitable"
		+	\|-
		+	\| colspan="5" align="center" \| '''Estimated worldwide human allele frequencies of ApoE '''
		+	\|-
		+	\| Allele \|\| ε2 \|\|ε3 \|\|ε4
		+	\|-
		+	\| General Frequency\|\|8\|\|78\|\|14
		+	\|-
		+	\| AD Frequency\|\|4\|\|59\|\|37
		+	\|}

	===Butterfly Hotspots===		===Butterfly Hotspots===

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 <hr>
 ==[[EBook_Problems_Contingency_Fit|Problems]]==
 <hr>

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 <center>[[Image:SOCR_EBook_Dinov_ChiSquare_042908_Fig2.jpg|500px]]</center>
-===Iris sepal and petal length==
+===Iris sepal and petal length===
 The [[SOCR_Data_052511_IrisSepalPetalClasses |Fisher's multivariate dataset on iris sepal and petal length]] provides another interesting example where we can look for how close are the petal or the sepal lengths of iris plant of different types.