A Trivariate Weibull Model for Oestradiol Plus Progestrone Treatment Increase...IOSRJM
In the previous studies a hypothesis was developed from the results by using trivariate Weibull model that the increase in leptin concentrations during the second half of the menstrual cycle may be related to changes in the steroidal milieu during the pre ovulatory period and the luteal phase. The present study was undertaken to test this hypothesis further by examining the effect of treatment with oestradiol and progesterone on leptine concentrations in normal pre menopausal women. The trivariate Weibull model is used for finding survival functions and log-likelihood functions for corresponding values of oestradiol, LH and Leptin for both untreated and treated with oestradiol and oestradiol plus progesterone respectively.
A Trivariate Weibull Model for Oestradiol Plus Progestrone Treatment Increase...IOSRJM
In the previous studies a hypothesis was developed from the results by using trivariate Weibull model that the increase in leptin concentrations during the second half of the menstrual cycle may be related to changes in the steroidal milieu during the pre ovulatory period and the luteal phase. The present study was undertaken to test this hypothesis further by examining the effect of treatment with oestradiol and progesterone on leptine concentrations in normal pre menopausal women. The trivariate Weibull model is used for finding survival functions and log-likelihood functions for corresponding values of oestradiol, LH and Leptin for both untreated and treated with oestradiol and oestradiol plus progesterone respectively.
Aligning Benchmarks With High Stakes Assessments 2009dvodicka
Overview of descriptive and inferential options for evaluating alignment of internal and external assessments to help improve student achievement.
Presented at Data Director User Conference in November 2009.
Basic Business Statistics Chapter 3Numerical Descriptive Measures
Chapters Objectives:
Learn about Measures of Center.
How to calculate mean, median and midrange
Learn about Measures of Spread
Learn how to calculate Standard Deviation, IQR and Range
Learn about 5 number summaries
Coefficient of Correlation
6MODULE 2Module 2 Problem SetEXAMPLEGrand .docxblondellchancy
6
MODULE 2
Module 2 Problem Set
EXAMPLE
Grand Canyon University: RES-866
Approaches to Research Design and Data Analysis
May 15, 2019
Running head: ASSIGNMENT TITLE HERE
1
Running head: MODULE 2
Introduction
Understanding and interpreting data is a vital component to a learner’s research as it reflects the study results. Properly analyzing the data is also crucial as this portrays the outcomes and delivers research findings which is the purpose of completing the study initially. Once data is extracted and analyzed, further development of the topic is discussed and shared allowing for a deeper sense of knowledge of the included variable, perceptions, and anything discovered throughout the study. Astroth and Chung (2018) discussed the importance of reviewing quantitative research studies highlighting the importance of ensuring the results are presenting and interpreted correctly. This article is geared towards nurses as they utilize evidence-based practices throughout many health care settings therefore properly analyzing data and accurately reporting results is critical in the potential care provided. For the purpose of this paper, the learner completed the assigned tasks of Module 2 Problem set which is provided below.
Drinks | Learning Activity 2.11
1. Create standardized scores for all scale variables (price through alcohol).
[DataSet1] \\Client\C$\Users\lauren.hazeltine\Downloads\Drinks.sav
Descriptive Statistics
N
Minimum
Maximum
Mean
Std. Deviation
Price per 6-pack
35
1.59
7.19
3.0274
1.12343
Cost per 12 Fluid Ounces
35
.27
1.20
.5057
.18732
Calories per 12 Fluid Ounces
35
68
175
139.77
24.447
Sodium per 12 Fluid Ounces in mg
35
6
27
14.66
6.145
Alcohol by Volume (in %)
35
2.30
5.50
4.5771
.60298
Valid N (listwise)
35
a. Which beverages have positive standardized scores on every variable?
· The beverages including each variable with positive standardized scores includes UA, UH, UL, UR, and SA.
b. What does this mean?
· Based on the raw data, these beverages are at or above the mean average or group mean
2. (a) What is the most extreme z-score on each variable? (b) What is the most extreme z-score across all variables?
a.
Variable
Product
Z-Score
price
SA
3.70524
cost
SA
3.70636
calories
UNR
-2.93581
sodium
PF & PJ
2.00859
alcohol
UNR
-3.77651
b.
cost
SA
3.70636
3. What beverage is most typical of all beverages, that is, has z-score values closest to 0 for these variables?
a. The most common beverage is UIR having the closest z-score value to 0.
4. (a) If the variable is normally distributed, what percentage of cases should be above 1 standard deviation from the mean or below 1 standard deviation from the mean?
a. 68%
(b) Calculate this percentage for a couple of the variables. Is the percentage of beverages with an absolute z-score above 1 close to the theoretical value?
b. Upon running calculations for some of the variables, the percentages were above 1 with an absolute z-score and cl ...
Aligning Benchmarks With High Stakes Assessments 2009dvodicka
Overview of descriptive and inferential options for evaluating alignment of internal and external assessments to help improve student achievement.
Presented at Data Director User Conference in November 2009.
Basic Business Statistics Chapter 3Numerical Descriptive Measures
Chapters Objectives:
Learn about Measures of Center.
How to calculate mean, median and midrange
Learn about Measures of Spread
Learn how to calculate Standard Deviation, IQR and Range
Learn about 5 number summaries
Coefficient of Correlation
6MODULE 2Module 2 Problem SetEXAMPLEGrand .docxblondellchancy
6
MODULE 2
Module 2 Problem Set
EXAMPLE
Grand Canyon University: RES-866
Approaches to Research Design and Data Analysis
May 15, 2019
Running head: ASSIGNMENT TITLE HERE
1
Running head: MODULE 2
Introduction
Understanding and interpreting data is a vital component to a learner’s research as it reflects the study results. Properly analyzing the data is also crucial as this portrays the outcomes and delivers research findings which is the purpose of completing the study initially. Once data is extracted and analyzed, further development of the topic is discussed and shared allowing for a deeper sense of knowledge of the included variable, perceptions, and anything discovered throughout the study. Astroth and Chung (2018) discussed the importance of reviewing quantitative research studies highlighting the importance of ensuring the results are presenting and interpreted correctly. This article is geared towards nurses as they utilize evidence-based practices throughout many health care settings therefore properly analyzing data and accurately reporting results is critical in the potential care provided. For the purpose of this paper, the learner completed the assigned tasks of Module 2 Problem set which is provided below.
Drinks | Learning Activity 2.11
1. Create standardized scores for all scale variables (price through alcohol).
[DataSet1] \\Client\C$\Users\lauren.hazeltine\Downloads\Drinks.sav
Descriptive Statistics
N
Minimum
Maximum
Mean
Std. Deviation
Price per 6-pack
35
1.59
7.19
3.0274
1.12343
Cost per 12 Fluid Ounces
35
.27
1.20
.5057
.18732
Calories per 12 Fluid Ounces
35
68
175
139.77
24.447
Sodium per 12 Fluid Ounces in mg
35
6
27
14.66
6.145
Alcohol by Volume (in %)
35
2.30
5.50
4.5771
.60298
Valid N (listwise)
35
a. Which beverages have positive standardized scores on every variable?
· The beverages including each variable with positive standardized scores includes UA, UH, UL, UR, and SA.
b. What does this mean?
· Based on the raw data, these beverages are at or above the mean average or group mean
2. (a) What is the most extreme z-score on each variable? (b) What is the most extreme z-score across all variables?
a.
Variable
Product
Z-Score
price
SA
3.70524
cost
SA
3.70636
calories
UNR
-2.93581
sodium
PF & PJ
2.00859
alcohol
UNR
-3.77651
b.
cost
SA
3.70636
3. What beverage is most typical of all beverages, that is, has z-score values closest to 0 for these variables?
a. The most common beverage is UIR having the closest z-score value to 0.
4. (a) If the variable is normally distributed, what percentage of cases should be above 1 standard deviation from the mean or below 1 standard deviation from the mean?
a. 68%
(b) Calculate this percentage for a couple of the variables. Is the percentage of beverages with an absolute z-score above 1 close to the theoretical value?
b. Upon running calculations for some of the variables, the percentages were above 1 with an absolute z-score and cl ...
dkNET Webinar: A New Approach to the Study of Energy Balance and Obesity usin...dkNET
Abstract
We report a web-based tool for analysis of experiments using indirect calorimetry to measure physiological energy balance. CalR simplifies the process to import raw data files, generate plots, and determine the most appropriate statistical tests for interpretation. Analysis using the generalized linear model (which includes ANOVA and ANCOVA) allows for flexibility in interpreting diverse experimental designs, including those of obesity and thermogenesis. Users also may produce standardized output files for an experiment that can be shared and subsequently re-evaluated using CalR. This framework will provide the transparency necessary to enhance consistency, rigor, and reproducibility. The CalR analysis software will greatly increase the speed and efficiency with which metabolic experiments can be organized, analyzed per accepted norms, and reproduced and has become a standard tool for the field. CalR is accessible at https://CalRapp.org/
The top 4 key questions that our tool can answer:
1. Can I reproducibly and transparently analyze indirect calorimetry experiments in under 10 minutes?
2. How hard is it to use Analysis of Covariance (ANCOVA) to determine whether my groups of animals are significantly different?
3. Is there an automated, reproducible way to exclude “noisy” outlier data from our indirect calorimetry experiments?
4. What are the key factors in determining metabolic rate of mice?
Presenter: Alexander Banks, PhD, principal investigator and assistant professor at Harvard Medical School and the Beth Israel Deaconess Medical Center.
dkNET Webinar Information: https://dknet.org/about/webinar
Stat170 - Introductory Statistics
Semester 2, 2015 Assignment 2
Instructions:
1. Type your answers directly into this document.
2. The answers to all questions are to be word processed. You can either type formulae into your solution or you can use the equation editor in Word or you may include hand-written equations and diagrams by photographing them so that you have the image saved as a picture file and then pasting (inserting) the image/s into your solution.
3. Your assignment should be uploaded as a Word (.doc or .docx) or PDF file (created from a Word processed document) ONLY. Other formats, including a PDF created from an image, will not be accepted by the system.
Information:
Q1
Q2
Q3
Q4
Total
10
13
10
17
50
1) Question 1 (10 marks)
Include an appropriate diagram for each part of this question. You can sketch these diagrams and paste in a photo of your sketch, along with your solutions.
a. BranCrunch is a new breakfast cereal. Boxes of BranCrunch are labelled ‘ 675 grams’ but there is some variation. The actual mean weight is 675 grams with a standard deviation of 21 grams.
i. Dan’s Discount Store sells BranCrunch in mega-packs of 8 boxes. Assuming the weights of boxes of BranCrunch are normally distributed, find the probability that the average weight of a mega-pack of BranCrunch is higher than 665 grams.
ii. Louie’s Convenience Store receives a shipment of 30 boxes of BranCrunch. Louie’s will complain to the manufacturers if the total weight of this shipment is lower than 20 kg. Find the probability that Louie’s Convenience Store will complain about the shipment and explain why the information about the weights of BranCrunch following a normal distribution is not necessary to answer this part of the question.
b. In 2014 the Department of Social Services reported that 32% of current marriages in Australia were expected to end in divorce.
Find the probability that more than 8 marriages out of a random sample of 20 marriages which were current in 2014 would end in divorce.
Question 2 (13 marks)
Lean body mass is the amount of weight carried on the body that is not fat. Metabolic rate is the rate at which the body consumes energy. The following Minitab output was constructed using data recorded in a fitness study which was designed to compare the lean body masses and also the metabolic rates of adolescent males and adolescent females. Use this output to answer the questions which follow.
Two-sample T for LeanBodyMass
Sex N Mean StDev SE Mean
Male 75 52.63 6.66 0.77
Female 75 43.42 6.05 0.70
Difference = μ (Male) - μ (Female)
Estimate for difference: 9.21
95% CI for difference: (****, ****)
T-Test of difference = 0 (vs ≠): T-Value = **** P-Value = **** DF = ****
Both use Pooled StDev = 6.3633
Two-sample T for MetRate
SE
Sex N Mean StDev Mean
Male 75 1626 227 26
Female 75 1258 172.
Presentation of EMPOWERING project in the last Workshop of the IEA Annex 58
poster draft 2
1. A Statistical Analysis on the Nutritional Intakes of Secondary School Children
An assessment of the impact of revised school food standards
Adverse outcomes of obesity include cardiovascular disease,
many cancers, type II diabetes, strokes, high blood pressure,
osteoarthritis, fertility problems, reduced life expectancy,
depression, anxiety and low self-esteem.
In 2002, 21.8% of boys and 27.5% of girls aged 2-15 years
were overweight or obese. Furthermore, the direct cost of
obesity to the NHS was estimated at £46-49 million per year.
In response to Jamie Oliver’s Feed Me Better campaign in
2005, the Department for Education and Skills revised the
national school food standards.
513 schoolchildren from 2 time-points (2000 and 2009)
completed ‘food diaries.’ From these, nutritionists devised
each child’s mean daily intake and mean lunchtime intake
for each nutrient (energy, protein, fat etc.).
Aim: Assess impact of standards
Variables that affect food/nutrient intake are:
• YEAR: 2000 or 2009: since changes were made to school
food regulations in this time period.
• LUNCH TYPE: School lunch (SL) or packed lunch (PL): since
regulations applied to school lunches only.
• SEX: Male or Female: since boys eat more than girls.
However, the difference between sexes does not depend
on the new standards and so this effect is not of interest.
The mean lunchtime energy intake in kcal:
Inference: Average energy intake decreased substantially for
school lunches, but not a lot for packed lunches.
Problem: The 4 groups do not contain equal amounts of boys
and girls, and since sex affects energy intake, the year/lunch
effects are confounded with the sex effect which is not of
interest. Therefore the groups are not comparable.
Solution: Adjusted means.
2000 2009 Difference: 2009-2000
SL 711.9 495.9 -216.0
PL 612.3 574.2 -38.2
Method:
1. Fit a linear regression model to the data:
𝑌𝑖𝑗𝑘𝓁 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + (𝛼𝛽)𝑖𝑗+𝛾 𝑘 + 𝜖𝑖𝑗𝑘𝓁 ,
* If the p-value for the interaction is significant, year affects intake differently for each lunch type, so a two-way table is needed to
present means. If however it is not significant, the interaction complicates the presentation, yet does not add anything worthwhile.
Therefore, the model will be re-fitted without the interaction if not significant and one-way tables used.
† Choice of 𝑆𝑒𝑥 is arbitrary as it does not affect the differences, but one that produces plausible mean values is preferable, so that
practitioners without statistics backgrounds are not disconcerted.
Response for 𝓁 𝑡ℎ subject,
who was from 𝑖 𝑡ℎ year,
𝑗 𝑡ℎ lunch type and 𝑘 𝑡ℎ sex
Overall mean
Effect of
𝑖 𝑡ℎ year
Effect of 𝑗 𝑡ℎ
lunch type
Effect of (𝑖𝑗) 𝑡ℎ
combination of
year and lunch
type*
Effect of 𝑘 𝑡ℎ sex
- to be corrected for
Error of the
𝓁 𝑡ℎ individual
2. Estimate regression coefficients & obtain equation for the fitted mean of each group:
𝑌𝑖𝑗 = 734.5 − 219.7 𝐼 𝑌𝑒𝑎𝑟 = 2009 − 106.4 𝐼 𝐿𝑢𝑛𝑐ℎ = 𝑆𝐿 − 39.1 𝑆𝑒𝑥𝑖𝑗 + 186.1 𝐼[𝑌𝑒𝑎𝑟 = 2009 & 𝐿𝑢𝑛𝑐ℎ = 𝑆𝐿]
where 𝐼[𝐴] is an indicator variable that equals 1 if the event A is true and 0 otherwise,
and 𝑆𝑒𝑥𝑖𝑗 is the proportion of females in the group.
3. Fix sex variable at a constant arbitrary† value, say the mean sex value of the sample:
S𝑒𝑥 = 0.5185
4. Compute the mean for each group at this uniform sex value, instead of using 𝑆𝑒𝑥𝑖𝑗
2000 school lunch: 𝑌0,0 = 734.5 − 219.7 × 0 − 106.4 × 0 − 39.9𝑆𝑒𝑥 + 186.1 × 0 × 0 = 713.8
2000 packed lunch: 𝑌0,1 = 734.5 − 219.7 × 0 − 106.4 × 1 − 39.9𝑆𝑒𝑥 + 186.1 × 0 × 1 = 607.5
2009 school lunch: 𝑌1,0 = 734.5 − 219.7 × 1 − 106.4 × 0 − 39.9𝑆𝑒𝑥 + 186.1 × 1 × 0 = 494.1
2009 packed lunch: 𝑌1,1 = 734.5 − 219.7 × 1 − 106.4 × 1 − 39.9𝑆𝑒𝑥 + 186.1 × 1 × 1 = 573.9
5. The group means have been adjusted for sex imbalance so they are comparable! Inference can now be made
on the differences (estimable quantities). This is because the differences are independent of choice of 𝑆𝑒𝑥
(when one mean is subtracted from another), 𝑆𝑒𝑥 cancels out – so differences are unique!
A package called lsmeans can be downloaded in R, allowing efficient calculation of adjusted group means, for
lunchtime and daily intakes of all nutrients. This package, by default, uses 0.5 for the arbitrary fixed value of 𝑆𝑒𝑥.
Diagnostic checks must be performed for each model, to check for homoscedasticity (constant variance, by residual
plots) and Normality (by Normal probability plots) of the estimated residuals.
For most models, the plots are satisfactory. However, lunchtime and daily vitamin C intake have concerning Normal
probability plots. The obvious curvature means that Normality cannot be assumed.
Normal probability plots for lunchtime and daily vitamin C intake
Problem: Significance tests and
confidence intervals are invalidated.
Solution: Data transformation:
a transformation must not change
the order of values, but can alter
the distance between successive
points to modify the overall shape
of the distribution and achieve a
‘bell curve’.
The Box-Cox power transformation (1964) is the most commonly used tool to remedy the
breakdown of the Normality assumption. For some positive data 𝑌1,…, 𝑌𝑛, it is given by
𝑌𝑖
(𝜆)
=
𝑌𝑖
𝜆
− 1
𝜆
, 𝑖𝑓 𝜆 ≠ 0,
log 𝑌𝑖 , 𝑖𝑓 𝜆 = 0,
where the transformation parameter 𝜆 requires estimation.
For non-positive data, there is a two-parameter version, which allows for a shift before
transformation, given by
𝑌𝑖
(𝜆)
=
(𝑌𝑖 + 𝜆2) 𝜆1−1
𝜆1
, 𝑖𝑓 𝜆1 ≠ 0,
log 𝑌𝑖 + 𝜆2 , 𝑖𝑓 𝜆1 = 0,
where the transformation parameter 𝜆1 and the shift parameter 𝜆2 both require estimation.
The Box-Cox parameters are usually estimated by maximum likelihood and then rounded to
resemble a practical transformation (e.g. square root, cube root, inverse).
The lunchtime vitamin C data is non-positive (contains some zeroes), so the two-parameter
version is used, giving 𝜆1 = 0.4022 and 𝜆2 = 0.0015 to 4 d.p. Rounding gives a square-root
transformation (preceded by a shift of size zero). The daily vitamin C data is positive. The
standard version is therefore used, giving 𝜆 = 0.1997. Rounding gives a log transformation.
Problem: Transformed data will typically be on a scale that is unfamiliar to practitioners.
Solution: Use the inverse transformation to back-transform the results, so that they are put on
the original scale and made accessible to practitioners.
Fitting the model to the square-rooted data gives an acceptable fit to Normality. The adjusted
means for the square-rooted data are calculated, then squared to convert them back to the
original scale:
Problem: It has been made apparent already that data on the original scale violates the
Normality assumption, which is the reason a transformation was sought in the first place.
Confidence intervals for the difference therefore cannot be found. Valid conclusions can only be
drawn from data on the square-root scale, which makes the back-transformation redundant.
Solution: Use log-transformation.
2000 2009 Difference 95% CI of difference
Square-root scale 𝑌2000 𝑌2009 𝑑 = 𝑌2009 − 𝑌2000
𝑑 ± 𝑠. 𝑒. 𝑑
Original scale
𝑌2000
2
𝑌2009
2
𝑌2009
2
- 𝑌2000
2
SL PL Difference 95% CI of difference
Square-root
scale
𝑌𝑆𝐿 𝑌𝑃𝐿 𝑑 = 𝑌𝑃𝐿 − 𝑌𝑆𝐿
𝑑 ± 𝑠. 𝑒. 𝑑
Original scale
𝑌𝑆𝐿
2
𝑌𝑃𝐿
2
𝑌𝑃𝐿
2
− 𝑌𝑆𝐿
2
A useful quality of the log-transformation is that an intuitive interpretation is possible upon back-
transformation. This is owing to the relationship between the geometric and arithmetic means of some
general data 𝑌1, 𝑌2, … , 𝑌𝑛
𝐺𝑀(𝑌𝑖) =
𝑖=1
𝑛
𝑌𝑖
1
𝑛
= exp
1
𝑛
𝑖=1
𝑛
log 𝑌𝑖 = exp 𝐴𝑀(log(𝑌𝑖) ,
where 𝐺𝑀(. ) and 𝐴𝑀(. ) denote the geometric and arithmetic means respectively. Therefore,
𝐴𝑀(log(𝑌𝑖)) = log 𝐺𝑀 𝑌𝑖 .
Hence, the difference between two group (arithmetic) means (of logged data) is given by
log 𝐺𝑀 𝑌𝐺𝑅𝑂𝑈𝑃 𝐴 − log 𝐺𝑀 𝑌𝐺𝑅𝑂𝑈𝑃 𝐵 = log
𝐺𝑀 𝑌𝐺𝑅𝑂𝑈𝑃 𝐴
𝐺𝑀 𝑌𝐺𝑅𝑂𝑈𝑃 𝐵
.
Upon exponentiation, the ‘difference’ simply becomes the ratio of the geometric means. Then, due to the
asymmetry of the log-transformation, the confidence interval of this ratio can be found directly by anti-
logging the confidence interval of the difference.
Unlike the square-root transformation however, a log-transformation cannot be applied to the
zero observations. This is resolved by shifting the data, but a sensible constant must be
determined.
Whichever one minimizes the residual
skewness is a logical choice, since
Normality corresponds to zero residual
skewness. Minimal residual skewness is
achieved with a shift of approximately
15. The log-transformation can then be
performed on the shifted intakes.
After log-transformation, there is still an acceptable fit to Normality. So although the Box-Cox method
indicated square-root, the log-transformation also manages to Normalize the data quite well.
Inference: Vitamin C intake in 2009 was 1.12 times larger than in 2000 and packed lunches on average
contained 1.07 times as much as school lunches.
2000 2009 Difference/Ratio 95% CI of difference
Log-scale 3.638 3.756 PL – SL: 0.118 (0.023, 0.212)
Original scale 38.0 42.8 PL / SL: 1.12 (1.02, 1.24)
SL PL Difference/Ratio 95% CI of difference
Log scale 3.664 3.730 PL – SL: 0.0664 (-0.0299, 0.1626)
Original scale 39.0 41.7 PL / SL: 1.07 (0.97, 1.18)
Lunchtime intakes: Consumption of energy, sodium and saturated fat declined significantly in school lunch
children, but not in packed lunch children. Vitamin C intake increased reasonably over the years, but the
impact was the same in both lunch types.
Daily intakes: Daily consumption of all nutrients did not differ for school and packed lunch children.
Consumption of energy and sodium fell significantly, but there was no evidence to suggest the same for
saturated fat. Vitamin C increased quite reasonably.
Problem: Energy intake is a proxy for amount eaten.
Energy intake decreased over the years, meaning that
children ate less in 2009 than in 2000. What if the
decrease in sodium is simply due to the fact that
they ate less food overall?
Solution: Investigate sodium-density.
To investigate how heavily sodium depends on energy, energy is included as an explanatory variable:
𝑁𝑎 = 𝛼 + 𝛽𝐸 + 𝜖,
where 𝑁𝑎 = daily sodium intake, 𝐸 = daily energy intake, 𝜖 = error, with 𝜖~𝑁(0, 𝜎2
), and 𝛼 incorporates the
effects of all other covariates as well as the general mean.
This time, means are not only adjusted for sex, but also for energy intake.
Hence, even if a child’s energy intake was the same in each year, their average daily sodium intake will still
have decreased by over 170 mg, which is a relatively large amount, suggesting that a reasonable amount of
the Na reduction is not attributed to reduced energy intake. So there has been a reduction in sodium-density.
Overall conclusions: standards have had a positive impact on school children’s diets, particularly in terms of
energy, sodium and sodium density.
2000 2009 Difference 95% CI of difference
Daily Na intake 2497.5 2323.8 -173.7 (-254.2, -93.2)
1. The childhood obesity crisis
2. Revised school food standards – a
response to the rising obesity levels
3. Project objective: were the
standards successful?
4. Factors affecting food intake
5. Simple analysis of lunchtime energy
intake
6. Adjusted means
7. Lsmeans
8. Diagnostic checks
9. Box-Cox power transformation
10. Square-root transformation of lunchtime Vit C intake
11. Shifting the lunchtime Vit C intakes
12. Unique property of log transformation
13. Log transformation of shifted Vit C intake
14. Summary of results