MATH 1015: Life Science Statistics. Lecture Pack for Chapter 1 Weeks 1-3. Lecturer: Jennifer Chan Room: Carslaw Room 817 Telephone:

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1 MATH 1015: Life Science Statistics Lecture Pack for Chapter 1 Weeks 1-3. Lecturer: Jennifer Chan Room: Carslaw Room 817 Telephone: Text: Phipps, M. and Quine, M. (2001) A Primer of Statistics (4th Ed.)

2 Lecture 1 Introduction and plots 1 Introduction 1.1 Statistics Statistics is a scientific study of numerical data based on natural phenomena. It is also the science of collecting, organising, interpreting and reporting data. There are four phases of an experiment, survey or study. 1. Planning Advise on the best way to collect data, what readings should be taken, how bias can be reduced or eliminated. This stage is guided by the questions the study wants to address, time and costs. 2. Data Analysis Numerical and graphical summaries of the data to get an impression of the variability and shape of the data. 3. Model Building Develop a mathematical model based on probability theory to explain the patterns observed. 4. Inference Use the model to make inferences about the population from which the sample was drawn. SydU MATH1015 (2007) First semester Dr. J. Chan 2

3 Lecture 1 Introduction and plots 1.2 Typical Statistical Problems 1. Quality Control Problem. In quality control, people regularly sample output and test for defects. Assume the process is deemed to be okay if the proportion of defective articles overall is at most 5%. The company inspects a sample of 60 articles and finds 4 defective articles. Should they reject the whole batch? 5% of 60 is 3. If the true defect rate was 5% can the observed 4 be explained by chance? We draw inferences about the population on the basis of a sample. 2. Polling data. - ACNeilsen Poll (SMH 1/03/03) Why choose a sample of size 1007? What is the margin for error and how is it calculated? 3. In vitro fertilization (IVF) data 6 of the first 7 births on the IVF program in Australia were female. Does this provide evidence of sex bias in IVF babies? SydU MATH1015 (2007) First semester Dr. J. Chan 3

4 Joke Lecture 1 Introduction and plots A Physicist, a Biologist, and a Statistician see two people enter a house, and then after some time, they see three people leave the house. The Physicist concludes, My initial observation must have been incorrect. The Biologist concludes, Clearly, the two reproduced.. The Statistician concludes, Well, if one more person enters the house, then there will be no-one in the house! SydU MATH1015 (2007) First semester Dr. J. Chan 4

5 Lecture 1 Introduction and plots 1.3 Data analysis Discrete data: Stem and Leaf diagrams; Frequency Tables. R code: > stem(x) Examples 1. No. of persons killed in road accidents in Local Govt areas of Sydney in SydU MATH1015 (2007) First semester Dr. J. Chan 5

6 Lecture 1 Introduction and plots 0 1,2,2,2,2,3,3,3,3,4,6,6,7,7,7,7,7,7,8,9,9,9,9 1 0,1,2,3,4,4,5,7,7,8 2 1,1,2,3,4 3 1 Double stem version: Stem Leaf 0 1,2,2,2,2,3,3,3,3,4 0 6,6,7,7,7,7,7,7,8,9,9,9,9 1 0,1,2,3,4,4 1 5,7,7,8 2 1,1,2,3, Stem and Leaf diagram gives an easy way to present ordered data. It gives an idea of the shape of the distribution of the data. 2. Yeast cell counts. (Data set 1, book P. 131.) i Total f i Solution: > x=c(2,2,4,... ) > tab=table(x) > tab x > stem(x) SydU MATH1015 (2007) First semester Dr. J. Chan 6

7 Lecture 1 Introduction and plots The decimal point is at the (Student s weight) Weight in pounds from 92 students: Males Females Construct a stem and leaf plot for this data set. Solution: SydU MATH1015 (2007) First semester Dr. J. Chan 7

8 Lecture 1 Introduction and plots Stem Leaf SydU MATH1015 (2007) First semester Dr. J. Chan 8

9 Lecture 1 Introduction and plots Continuous data: Histograms and Line plots Unlike counts, measurements are often rounded. Be aware of the rounding when summarising the data set. Age is generally rounded down whereas lengths are rounded to the nearest value. A small discrete and continuous data set can be summarized visually using a line plot. Frequency tables give the frequencies for various intervals. - Usually use intervals. - Take care in determining the true class boundaries Relative frequency is the interval frequency divided by the total number of observations. Histograms - Column sides are placed at the interval boundaries. - The area of the column reflects the frequency for the interval. - Unequal class intervals are used to emphasise important structure in the data. - R code: > hist(x) SydU MATH1015 (2007) First semester Dr. J. Chan 9

10 Lecture 1 Introduction and plots Example: (Student s weight) Construct a frequency table, a histogram and a line plot for this data set. Solution: The min and max values are 95 and 215 respectively. The range is = 120. For 8 classes, the class width is 120/8 = 15. CLASS INTERVAL MIDPOINT FREQUENCY RELATIVE FREQUENCY TOTAL Relative freq. Frequency Histogram and line plot SydU MATH1015 (2007) First semester Dr. J. Chan 10

11 Lecture 1 Introduction and plots Histogram with unequal class intervals. Example: (Cervical cancer data) Data set 4, book P.132. Age Frequency Before drawing a histogram calculate the frequency of cancer per year. This gives the height of the rectangles. Age Boundaries Length Freq. Freq/year [20,30) /10= [30,35) /5= [35,40) /5= [40,55) /15= [55,60) /5= [60,70) /10= [70,90) /20=0.65 Freq./year Histogram SydU MATH1015 (2007) First semester Dr. J. Chan 11

12 Lecture 2 5-nos. summaries & boxplot 2 5-number summaries and boxplots number summaries Measures of centre: Median, mean, mode. The median, range and IQR are easily calculated from an ordered list of the data. Median The median, x is a value such that at least half the observations are less than or equal to x and at least half the observations are greater than or equal to x. To find the median, we arrange the data in ascending order. If the number of data is ODD the median is the middle data point. If the number of data is EVEN, we average the 2 values around the middle: = The median Middle space Measures of spread: range, interquartile range (IQR), standard deviation. Two data sets with the same center measures may have completely different spreads. The measures of center and spread taken together give a better picture of the shape of a data set. SydU MATH1015 (2007) First semester Dr. J. Chan 12

13 Lecture 2 5-nos. summaries & boxplot Range The range = Maximum value - Minimum value. Quartiles The lower quartile, Q 1, is a value such that at least 25% of the observations are less than or equal to Q 1 and at least 75% of the observations are greater than or equal to Q 1. Similarly the upper quartile, Q 3, separates off the upper 25% of the observations in an ordered list. Interquartile Range IQR = Q 3 Q 1. The middle 50% of the observations lie in [Q 1, Q 3 ]. 5-number summary 2.2 Boxplot Q 1 Q 2 Q 3 25% 25% 25% 25% IQR (Minimum, Q 1, x, Q 3, Maximum). It is a quick way of presenting this information graphically. Q 1 and Q 3 determine the ends of the box. Outliers are points more than one IQR beyond the ends of the box. To determine the outliers we calculate the upper and lower thresholds. SydU MATH1015 (2007) First semester Dr. J. Chan 13

14 Lecture 2 5-nos. summaries & boxplot IQR IQR LT UT = Q 1 IQR = Q 3 + IQR. Min 50% 50% 50% Outlier, max An example with min within LT and max beyond UT. Shapes of Data Sets Boxplots give an easy graphical means of getting an impression of the shape of the data set. The shape is used to suggest a mathematical model for the situation of interest. 1. Symmetric 2. Right skewed (positive skewness): the boxplot is stretched to the right. 3. Left skewed (negative skewness): the boxplot is stretched to the left. SydU MATH1015 (2007) First semester Dr. J. Chan 14

15 Lecture 2 5-nos. summaries & boxplot Transformation We can often find a simple transformation that will make the data more symmetric. For right skewed data log or square root transformations often work. R code: > s=summary(x) > s > IQR=s[5]-s[2] > IQR > UT=s[5]+IQR > UT > LT=s[2]-IQR > LT > range(x) > boxplot(x) SydU MATH1015 (2007) First semester Dr. J. Chan 15

16 Lecture 2 5-nos. summaries & boxplot Example: (Student weight) Find the min, max, Q 1, Q 3, IQR, LT and UT. Draw the boxplot. Solution: Min = 95 range = = 120 Max = 215 Median = X 0.5(92+1) = X 45.5 = X 45 + X 46 2 Q 1 = X 0.25(92+1) = X = X 23 + X 24 2 Q 3 = X 0.75(92+1) = X = X 69 + X 70 2 IQR = = 31 LT = Q 1 IQR = = 94 UT = Q 3 + IQR = = = = = = = = IQR 50% 125 IQR % 50% Min Outlier, max 215 Example: (Road Accident data) Number of persons killed in road accidents in Local Govt areas of Sydney in ,2,2,2,2,3,3,3,3,4,6,6,7,7,7,7,7,7,8,9,9,9,9 1 0,1,2,3,4,4,5,7,7,8 2 1,1,2,3,4 3 1 n = 39. Range = 31-1=30. SydU MATH1015 (2007) First semester Dr. J. Chan 16

17 Lecture 2 5-nos. summaries & boxplot x = X (39+1)/2 = X 20 = 9 Q 1 = X (39+1)/4 = X 10 = 4 Q 3 = X 3(39+1)/4 = X 30 = 15 IQR = 15 4 = 11. UT = = 26 < 31 and LT = 4 11 = 7 < 1. The max no. 31 is an outlier. > x=c(1,2,2,2,2,3,3,3,3,4,6,6,7,7,7,7,7,7,8,9,9,9,9,10,11,12,13,14,14,15, 17,17,18,21,21,22,23,24,31) > sqrtx=sqrt(x) > par(mfrow=c(2,2)) > boxplot(x) > title("x") > boxplot(sqrtx) > title("square root of x") R Output: x square root of x Are the data roughly symmetrical about the median, left or right skewed? It is right skewed and it becomes symmetric after taking square root transformation. SydU MATH1015 (2007) First semester Dr. J. Chan 17

18 Lecture 2 5-nos. summaries & boxplot 2.3 Estimating Quartiles Example: (Diameters) Data set 3, book P.132. Frequency Table Intervals Frequency Boundaries Cum Freq n = 200. Class width = We can approximate the quartiles from the cumulative frequency diagram. x = X (200+1)/2 = X = Q 1 = X (200+1)/4 = X = = = Q 3 = X 3(200+1)/4 = X = = To construct the cumulative frequency diagram plot the cumulative frequency against the class interval UPPER boundary. SydU MATH1015 (2007) First semester Dr. J. Chan 18

19 f ABC ADE x C 55 E A 28 D B ? Lecture 2 5-nos. summaries & boxplot x SydU MATH1015 (2007) First semester Dr. J. Chan 19

20 Lecture 3 Sample mean and variance 3 Sample mean and variance 3.1 Review of summation notation For the values x 1 = 3, x 2 = 4, x 3 = 5, x 4 = 3 evaluate the following summation expressions i=2 4 x i = = 15 x 2 i = = 59 x i = = 9 (2x i + 3) = = Sample Mean The sample mean is the simple average of the observations. observations x 1, x 2,..., x n For x = x 1 + x x n n If e i = cx i + f then ē = c x + f. n e i = c n x i + nf = 1 n n x i. SydU MATH1015 (2007) First semester Dr. J. Chan 20

21 Joke Lecture 3 Sample mean and variance Did you hear about the statistician who had his head in an oven and his feet in a bucket of ice? When asked how he felt, he replied, On the average I feel just fine. When she told me I was average, she was just being mean. Grouped frequency table. If we only have the information provided by a grouped frequency table, for example, we only have access to the published report and not the original data set, then we can approximate the sample mean by x = 1 n k (f i u i ), where the interval centres are u 1, u 2,..., u k with corresponding frequencies f 1, f 2,..., f k. Example: (Diameters) Data set 3, book P.132. Frequency Table Intervals Frequency (f j ) Interval centre (u j ) SydU MATH1015 (2007) First semester Dr. J. Chan 21

22 Lecture 3 Sample mean and variance There are k = 12 intervals. n = f i u i = 2(13.12) + 1(13.17) + 8(13.22) (13.67) = x = 1 n 12 f i u i = = The mean of the raw data was SydU MATH1015 (2007) First semester Dr. J. Chan 22

23 Lecture 3 Sample mean and variance 3.3 Mean vs Median 1. Mean is easier to calculate and easier to handle theoretically. 2. If the data are roughly symmetric then the mean, median, and mode are close and they lie at the center of the distribution. 3. If the data are skewed then the mean is pulled toward the long tail. We have mode median mean if it is right skewed. 4. The median is robust against outliers and incorrect readings whereas the mean is not. Example: (Heat of sublimation of platinum) Data set 14, book P.136. Stem and Leaf Display ,2,3,5,7,8,8,8,9, ,0,2,2,4,4,8, , SydU MATH1015 (2007) First semester Dr. J. Chan 23

24 Lecture 3 Sample mean and variance n = 26. Median: x = X (26+1)/2 = X 13.5 = 1 2 (X 13 + X 14 ) = 1 2 (0 + 2) = 1. In Data set 14, if is changed to 34.1 the median does not change but the sample mean changes. SydU MATH1015 (2007) First semester Dr. J. Chan 24

25 Lecture 3 Sample mean and variance 3.4 Sample variance and standard deviation. An alternative to the IQR as a measure of spread is the sample standard deviation, s. For data x 1, x 2,..., x n The sample variance is Calculation formula is s = 1 n 1 s 2 = 1 n 1 s 2 = 1 n 1 where S xx = n (x i x) 2 n n n (x i x) 2. (x i x) 2. x 2 i 1 n ( n x i ) 2 If we use working origin a and working units h with d i = x i a h then s x = hs d. For data from frequency tables we use s 2 = 1 n 1 k j=1 f j (u j ū) 2. SydU MATH1015 (2007) First semester Dr. J. Chan 25

26 Lecture 3 Sample mean and variance Example: Solution: n = 12. First calculate xi = = 689 x 2 i = = Mean: x = 1 n xi = = Variance: s 2 1 = x 2 i ( x i ) 2 = 1 n 1 n (689)2 12 = Standard Deviation: s = = Example: (Interobital width) A random sample of 12 measurements of interobital width of domestic pigeons is obtained as follow: Find the mean, median, mode, variance, standard deviation, range and quartiles. Construct a boxplot for these data. Solution: Arranged: mean = n x i n = = SydU MATH1015 (2007) First semester Dr. J. Chan 26

27 Lecture 3 Sample mean and variance median = = 11.8 mode = 11.8 and 12.2 not unique s 2 = 1 n 1 n x 2 i n x 2 = = s = = range = = 2.6 Q 1 = = Q 3 = = 12.2 IQR = = 1.15 (LT,UT) = ( , ) = (9.325, ) (no outliers) SydU MATH1015 (2007) First semester Dr. J. Chan 27

28 Lecture 4 Scatter plot and correlation 4 Scatter plot and correlation Procedures considered so far only involve observations on a single feature. Often we take several readings on each subject or experimental unit. For example, x y patient s age blood pressure temperature reaction time alcohol consumption cholesterol level. 4.1 Scatterplot The first step is to construct a scatterplot of the observed pairs. Example: (Weight & height) Height is frequently named as a good predictor for weight among people of the same gender. Give a scatterplot of the following heights (in cm) and weights (in kg) from 14 males between the ages of 19 and 26 years. Weight Height Solution: In R, > weight=c(83.9,99,63.8,...) > height=c(185,180,173,...) > par(mfrow=c(1,1)) > plot(weight,height) > title("weight against height") R output: (scatter plot) SydU MATH1015 (2007) First semester Dr. J. Chan 28

29 Lecture 4 Scatter plot and correlation Weight against height height weight 4.2 Correlation Coefficient The correlation coefficient is a numerical index that measures the degree of linear association between x and y. r = Σ n (x i x)(y i ȳ) Σ n (x i x) 2 Σ n (y i ȳ) 2 = S xy Sxx S yy, where S xx = n (x i x) 2 S yy = n (y i ȳ) 2 S xy = n (x i x)(y i ȳ). Note that if we rescale the x or the y values we do not change r. If we replace x i by w i = cx i + a for all i = 1, 2,.., n then w = c x + a and so (w i w) = c(x i x). SydU MATH1015 (2007) First semester Dr. J. Chan 29

30 Lecture 4 Scatter plot and correlation Consider n = n = n 2 Syy x i x y i ȳ Sxx ( xi x Sxx ) 2 2 n (x i x) 2 S xx + = S xx S xx + S yy S yy 2 n (y i ȳ) 2 ( ) xi x Sxx S yy 2 y i ȳ + n Syy n S xy Sxx S yy = 2 2r. 2 Syy y i ȳ (x i x)(y i ȳ) Sxx S yy Thus 2 2r 0 so 1 r. If r = 1 then x i x Sxx = y i ȳ Syy for all i. Thus the points (x i, y i ) all lie on the straight line y = mx + d, where m is the slope and d is the y-intercept. Similarly, n so 2 + 2r 0 so r 1. 2 Syy x i x + y i ȳ Sxx = 2 + 2r, 1 r 1. SydU MATH1015 (2007) First semester Dr. J. Chan 30

31 Lecture 4 Scatter plot and correlation If r = 1 then the observations (x i, y i ) all lie on a straight line with negative slope. Characteristics of r 1. Scale free r If r = ±1.0, then all the observations fall on a straight line. 4. Note x and y can have a very strong non-linear relationship and r = Remember a high correlation coefficient does not necessarily imply any causal relationship between the two variables. r is positive Y X r is negative Y X r is zero Y X Perfect fit r = 1, σ 2 = 0 Y X Imperfect fit r < 1, σ 2 > 0 Y X Perfect negative correlation No correlation Perfect positive correlation Strong negative Moderate negative Weak negative Weak positive Moderate positive Strong positive correlation correlation correlation correlation correlation correlation ve r +ve r 1.00 SydU MATH1015 (2007) First semester Dr. J. Chan 31

32 Lecture 4 Scatter plot and correlation Example (Weight and height) Calculate the correlation coefficient. Solution: In R: > cor(weight,height) [1] Example (Soil temp. and germination interval) Soil temperature (x i ) and germination interval (y i, in days) (book P.37) were observed for plots of winter wheat in 10 localities: x i y i Solution: n = 10 n x i = = 58.5 n y i = = 251 n x2 i = = n y2 i = = 6985 n x iy i = 12.5(10) + 5(26) (33) = S xx = n x 2 i n x 2 = ( ) = S yy = n yi 2 nȳ 2 = ( ) = SydU MATH1015 (2007) First semester Dr. J. Chan 32

33 Lecture 4 Scatter plot and correlation r = S xy = n x i y i n xȳ = (5.850)(925.1) = S xy Sxx S yy = (684.9) = SydU MATH1015 (2007) First semester Dr. J. Chan 33

34 Lecture 5 Regression 5 Regression Consider the weight and height data. How do we fit a trend line to a data set like this? Data: (x 1, y 1 ), (x 2, y 2 ),.., (x n, y n ) 5.1 Regression line Using a regression line y = a + bx the estimated value of y at x i is ŷ i = a + bx i. The observed value of y at x i is y i. The residual error at x i is e i = y i ŷ i. We minimise e 2 i = n (y i (a + bx i )) 2 to find a and b. b = S xy /S xx Note b = S xy Sxx S yy a = ȳ b x. S yy S xx = r S yy S xx. The slope has the same sign as the correlation coefficient. SydU MATH1015 (2007) First semester Dr. J. Chan 34

35 Lecture 5 Regression Example: (Weight & height) Fit a regression line ŷ = a + bx to the data. Solution: We have y i i yi 2 i = ; x i = 2472; i = ; x 2 i = ; i i x i y i = S yy y = x = = i i y i n i x i = 1, = n = = yi 2 ny 2 = = S xy S xx = i = i x i y i nxy = = x 2 i nx 2 = = Hence ˆb = â = S xy S xx = = , ȳ ˆb x = = , The regression line is Weight= *Height. When x = 0, ŷ = â = is an imaginary level of the predicted weight when the height is 0 which is impossible. For each 1 cm increase in height, the weight will be increased by ˆb = kg. In R, > y=weight SydU MATH1015 (2007) First semester Dr. J. Chan 35

36 Lecture 5 Regression > x=height > c=lsfit(x,y)$coeff > c Intercept X > par(mfrow=c(1,1)) > plot(x,y,xlab="height",ylab="weight") > abline(c[1],c[2]) > title("fitted line") R output: (fitted line plot) fitted line weight height 5.2 Residual Plots If the plot of residuals against x shows any strong structure then a more complex model is needed. SydU MATH1015 (2007) First semester Dr. J. Chan 36

37 Lecture 5 Regression A random scatterplot indicates the model assumtions are OK. e ŷ Nonconstant variance σ 2 e, σ 2 e increases with x. e ŷ Residuals plots e vs ŷ Functional form f(x) may be wrong. It should be f(x) = β 0 + β 1 x + β 2 x 2 + β 3 x 3. e ŷ y y y x x Fitted line Y vs x x Example: (Weight & height) What will be the average weight of those males who are 175 cm tall? Check the regression model using residual plot. Solution: In R, > predict=c[1]+c[2]*175 > predict Intercept > res=lsfit(x,y)$res > fitted=y-res > par(mfrow=c(2,2)) > plot(fitted,res) > abline(h=0) SydU MATH1015 (2007) First semester Dr. J. Chan 37

38 Lecture 5 Regression > title("residual plot") > boxplot(res) > title("boxplot of residuals") R output: (fitted line plot) residual plot boxplot of residuals res fitted The predict weight is kg. Apart from an outlier, the residual plot shows that the errors are random and symmetric. Example: (Dose & urine concentration) Dose x in gms and concentration in the urine, y (in mg/gm): x: y: n = 12 i x i = 507 x = = i y i = 144 y = = 12 i x 2 i = i y 2 i = 1802 SydU MATH1015 (2007) First semester Dr. J. Chan 38

39 Lecture 5 Regression i x i y i = 6314 S xx = i x 2 i nx 2 = = S yy = i y 2 i ny 2 = = 74 S xy = i x i y i nxy = = 230 Correlation coefficient: Regression: r = S xy Sxx S yy = b = S xy S xx = = , (74) = a = ȳ b x = = , Fitted line: Concentration= *dose. SydU MATH1015 (2007) First semester Dr. J. Chan 39

40 Lecture 5 Regression 5.3 Regression Effect 1. For large data sets split the x-axis into narrow strips and find the average of the y values in each strip of the scatterplot. 2. Means often scatter around a straight line called the regression line. The regression line reflects how the average y values vary with x. 3. In Galton s data the scatterplot is roughly elliptical. Average father s height x = 68 inches Average son s height ȳ = 69 inches r = Sons of tall fathers tend to be shorter than their fathers whereas sons of short fathers tend to be taller than their fathers on average. Galton noted the regression effect sometimes called regression to the mean. SydU MATH1015 (2007) First semester Dr. J. Chan 40