SMHS ModelFitting - Revision history

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

2014-10-21T13:19:20Z

‎Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test)

Clgalla: /* Theory */

2014-10-21T13:05:26Z

‎Theory

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

2014-10-21T12:55:33Z

‎Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test)

Clgalla at 20:27, 20 October 2014

2014-10-20T20:27:40Z

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

2014-10-20T20:24:06Z

‎Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test)

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

2014-10-20T19:43:24Z

‎Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test)

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Dinov: Created page with '== Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) ==
* SOCR Home page: http://www.socr.umich.edu {{translate|pageName=http…'

2014-06-30T20:46:33Z

Created page with '== Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) == <hr> * SOCR Home page: http://www.socr.umich.edu {{translate|pageName=http…'

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==[[SMHS| Scientific Methods for Health Sciences]] - Model Fitting and Model Quality (KS-test) ==

<hr>
* SOCR Home page: http://www.socr.umich.edu

{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SMHS_ModelFitting}}

@@ Line 83: / Line 83: @@
   sort.data <- sort(data)
   percentile <- c(0.167, 0.333, 0.5, 0.667, 0.8333)
-  plot(ecdf(data),verticals=T,xlim=c(-1.5,1.5),ylim=c(0,1), xlab='Data',   ylab='' ,main='Cumulative Fraction Plot vs. Percentil Plot')
+  plot(ecdf(data),verticals=T,xlim=c(-1.5,1.5),ylim=c(0,1), xlab='Data',
   par(new=T)
   plot(sort.data,percentile,type='o',xlim=c(-1.5,1.5),ylim=c(0,1),col=2,xlab='', ylab='', main='')

@@ Line 29: / Line 29: @@
 Note: two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is. While the K-S test is usually used to test whether a given $F(x)$ is the underlying probability distribution $F_{n}(x)$, the procedure may be inverted to give confidence limits on $F(x)$ itself. If we choose a critical value of the test statistics $D_{\alpha}$ such that $P(D_{n}>D_{\alpha})=\alpha$, then a band of width $\pm D_{\alpha}$ around $F_{n}(x)$ will entirely contain $F(x)$ with probability $1-\alpha.$
-) Illustration on how the K-S test works with example: consider the two datasets control$={1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95, 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38}$, treatment= ${2.37, 2.16, 14.82, 1.73, 41.04, 0.23, 1.32, 2.91, 39.41, 0.11, 27.44, 4.51, 0.51, 4.50, 0.18, 14.68, 4.66, 1.30, 2.06, 1.19}$.
+'''3) Illustration on how the K-S test works with example:''' consider the two datasets control$={1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95, 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38}$, treatment= ${2.37, 2.16, 14.82, 1.73, 41.04, 0.23, 1.32, 2.91, 39.41, 0.11, 27.44, 4.51, 0.51, 4.50, 0.18, 14.68, 4.66, 1.30, 2.06, 1.19}$.
 *Descriptive statistics: to reduce the list of all the data items to a few simpler numbers. For the control group, the mean is 3.607, the median is 0.60, the highest is 50.57, the lowest is 0.08 and the standard deviation is 11.165. Obviously, this dataset is not normally distributed.
 *Sort the control data: control sorted$={0.08, 0.10, 0.15, 0.17, 0.24, 0.34, 0.38, 0.42, 0.49, 0.50, 0.70, 0.94, 0.95, 1.26, 1.37, 1.55, 1.75, 3.20, 6.98, 50.57},$ evidently no data lies strictly below $0.08, 5% = 1/20$ of the data is smaller than $0.1$, 10% of the data is strictly smaller than $0.15, 15%$ of the data is strictly smaller than $0.17$, $\cdots.$ There are $17$ data points smaller than $\pi$, and we say that the cumulative fraction of the data smaller than $\pi$ is $17/20 (0.85).$
@@ Line 76: / Line 76: @@
   (ecdf(control))(1)-(ecdf(treatment))(1)   ## D=0.45  same as using the log-scale
-'''4) Using the Percentile Plot:''' for our habit of observing and comparing continuous curves. We may seek to use something similar to cumulative fraction plot, but without the odd steps, say, try the percentiles. Consider the dataset of ${-0.45, 1.11, 0.48, -0.82, -1.26}.$ Sort this data from small to large ${-1.26, -0.82, -0.45, 0.48, 1.11}.$ The median is $-0.45$, which is the 50th percentile. To calculate the percentile, denote the point’s location in the sorted dataset as r, and then divided by the number of points plus one: percentile = $r/(N+1)$. Now we have the set of (data, percentile) pairs: ${(-1.26, 0.167), (-0.82, 0.333), (-0.45, 0.5), (0.48, 0.667), (1.11, 0.833)}$. We can connect the adjacent data points with a straight line and the resulting collection of connected straight line segment is called an ''ogive'''''Bold text''').
+'''4) Using the Percentile Plot:''' for our habit of observing and comparing continuous curves. We may seek to use something similar to cumulative fraction plot, but without the odd steps, say, try the percentiles. Consider the dataset of ${-0.45, 1.11, 0.48, -0.82, -1.26}.$ Sort this data from small to large ${-1.26, -0.82, -0.45, 0.48, 1.11}.$ The median is $-0.45$, which is the 50th percentile. To calculate the percentile, denote the point’s location in the sorted dataset as r, and then divided by the number of points plus one: percentile = $r/(N+1)$. Now we have the set of (data, percentile) pairs: ${(-1.26, 0.167), (-0.82, 0.333), (-0.45, 0.5), (0.48, 0.667), (1.11, 0.833)}$. We can connect the adjacent data points with a straight line and the resulting collection of connected straight line segment is called an '''''ogive'''''.

@@ Line 48: / Line 48: @@
   plot(ecdf(control),verticals=TRUE,main='Cumulative Fraction Plot')
+'''5) The K-S statistic in more than one dimension:''' modifies the K-S test statistic to accommodate the multivariate data. Given that the maximum difference between two joint $cdf$ is not generally the same as the maximum difference of any of the complementary distribution functions, the modification is not straightforward in this way. Instead, the maximum difference will differ depending on which of $Pr(x<X \Lambda Y>y)$ or any of the other two possible arrangement is used. One approach may be to compare the cdfs of the two samples with all possible orderings, and take the largest as the K-S statistic. In $d$ dimensions, there are $2^{d}-1$ orderings. The critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.
 ===Applications===
@@ Line 146: / Line 112: @@
 ===Problems===
-.1) Consider the following dataset, control={0.22, -0.87, -2.39, -1.79, 0.37, -1.54, 1.28, -0.31, -0.74, 1.72, 0.38, -0.17, -0.62, -1.10, 0.30, 0.15, 2.30, 0.19, -0.50, -0.09}, treatment={-5.13, -2.19, -2.43, -3.83, 0.50, -3.25, 4.32, 1.63, 5.18, -0.43, 7.11, 4.87, -3.10, -5.81, 3.76, 6.31, 2.58, 0.07, 5.76, 3.50}, use the K-S test to test the equality of the two samples. Include all necessary steps and plots.
+) Consider the following dataset, control={0.22, -0.87, -2.39, -1.79, 0.37, -1.54, 1.28, -0.31, -0.74, 1.72, 0.38, -0.17, -0.62, -1.10, 0.30, 0.15, 2.30, 0.19, -0.50, -0.09}, treatment={-5.13, -2.19, -2.43, -3.83, 0.50, -3.25, 4.32, 1.63, 5.18, -0.43, 7.11, 4.87, -3.10, -5.81, 3.76, 6.31, 2.58, 0.07, 5.76, 3.50}, use the K-S test to test the equality of the two samples. Include all necessary steps and plots.
-.2) Revise the example given in the lecture with a different control group: control={1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95, 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38}, run the K-S test again and state clearly your conclusions with necessary plots.
+) Revise the example given in the lecture with a different control group: control={1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95, 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38}, run the K-S test again and state clearly your conclusions with necessary plots.
-.3) Two types of plants are in bloom in wood and we want to study if bees prefer one tree to the other? We collect data by using a stop-watch to time how long a bee stays near a particular tree. We begin to time when the bee touches the tree; we stop timing when the bee is more than a meter from the tree. (As a result all our times are at least 1 second long: it takes a touch-and-go bee that long to get one meter from the tree.) We wanted to time exactly the same number of bees for each tree, but it started to rain. (Unequal dataset size is not a problem for the KS-test.) Apply the K-S test with this example. The data are given as below:
+) Two types of plants are in bloom in wood and we want to study if bees prefer one tree to the other? We collect data by using a stop-watch to time how long a bee stays near a particular tree. We begin to time when the bee touches the tree; we stop timing when the bee is more than a meter from the tree. (As a result all our times are at least 1 second long: it takes a touch-and-go bee that long to get one meter from the tree.) We wanted to time exactly the same number of bees for each tree, but it started to rain. (Unequal dataset size is not a problem for the KS-test.) Apply the K-S test with this example. The data are given as below:
 T1={23.4, 30.9, 18.8, 23.0, 21.4, 1, 24.6, 23.8, 24.1, 18.7, 16.3, 20.3, 14.9, 35.4, 21.6, 21.2, 21.0, 15.0, 15.6, 24.0, 34.6, 40.9, 30.7, 24.5, 16.6, 1, 21.7, 1, 23.6, 1, 25.7, 19.3, 46.9, 23.3, 21.8, 33.3, 24.9, 24.4, 1, 19.8, 17.2, 21.5, 25.5, 23.3, 18.6, 22.0, 29.8, 33.3, 1, 21.3, 18.6, 26.8, 19.4, 21.1, 21.2, 20.5, 19.8, 26.3, 39.3, 21.4, 22.6, 1, 35.3, 7.0, 19.3, 21.3, 10.1, 20.2, 1, 36.2, 16.7, 21.1, 39.1, 19.9, 32.1, 23.1, 21.8, 30.4, 19.62, 15.5}
 T2={16.5, 1, 22.6, 25.3, 23.7, 1, 23.3, 23.9, 16.2, 23.0, 21.6, 10.8, 12.2, 23.6, 10.1, 24.4, 16.4, 11.7, 17.7, 34.3, 24.3, 18.7, 27.5, 25.8, 22.5, 14.2, 21.7, 1, 31.2, 13.8, 29.7, 23.1, 26.1, 25.1, 23.4, 21.7, 24.4, 13.2, 22.1, 26.7, 22.7, 1, 18.2, 28.7, 29.1, 27.4, 22.3, 13.2, 22.5, 25.0, 1, 6.6, 23.7, 23.5, 17.3, 24.6, 27.8, 29.7, 25.3, 19.9, 18.2, 26.2, 20.4, 23.3, 26.7, 26.0, 1, 25.1, 33.1, 35.0, 25.3, 23.6, 23.2, 20.2, 24.7, 22.6, 39.1, 26.5, 22.7}

@@ Line 34: / Line 34: @@
   RCODE:
-  Control <- c(1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95 , 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38)
+  Control <- c(1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15,
-  Treatment <- c(2.37, 2.16, 14.82, 1.73, 41.04, 0.23, 1.32, 2.91, 39.41, 0.11, 27.44, 4.51,   0.51, 4.50, 0.18, 14.68, 4.66, 1.30, 2.06, 1.19)
+.49, 0.95 , 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38)
   summary(control)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.

@@ Line 27: / Line 27: @@
 </center>
-Note: two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is. While the K-S test is usually used to test whether a given F(x) is the underlying probability distribution F_{n}(x), the procedure may be inverted to give confidence limits on F(x) itself. If we choose a critical value of the test statistics D_{\alpha} such that P(D_{n}>D_{\alpha})=\alpha, then a band of width \pm D_{\alpha} around F_{n}(x) will entirely contain F(x) with probability 1-\alpha.
+Note: two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is. While the K-S test is usually used to test whether a given $F(x)$ is the underlying probability distribution $F_{n}(x)$, the procedure may be inverted to give confidence limits on $F(x)$ itself. If we choose a critical value of the test statistics $D_{\alpha}$ such that $P(D_{n}>D_{\alpha})=\alpha$, then a band of width $\pm D_{\alpha}$ around $F_{n}(x)$ will entirely contain $F(x)$ with probability $1-\alpha.$
 ) Illustration on how the K-S test works with example: consider the two datasets control$={1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95, 0.24, 1.37, 0.17, 6.98, 0.10, 0.94, 0.38}$, treatment= ${2.37, 2.16, 14.82, 1.73, 41.04, 0.23, 1.32, 2.91, 39.41, 0.11, 27.44, 4.51, 0.51, 4.50, 0.18, 14.68, 4.66, 1.30, 2.06, 1.19}$.

SMHS ModelFitting - Revision history

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

Clgalla: /* Theory */

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

Clgalla at 20:27, 20 October 2014

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

Clgalla: /* Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) */

Dinov: Created page with '== Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) == * SOCR Home page: http://www.socr.umich.edu {{translate|pageName=http…'

Dinov: Created page with '== Scientific Methods for Health Sciences - Model Fitting and Model Quality (KS-test) ==
* SOCR Home page: http://www.socr.umich.edu {{translate|pageName=http…'