ChatGPT: Regression analysis makes several assumptions about the data and the relationship between the variables being studied. These assumptions include:
Linearity: The relationship between the independent and dependent variables is linear.
Independence: The observations are independent of each other.
Homoscedasticity: The variance of the errors is constant across all levels of the independent variable.
Normality: The errors are normally distributed.
No multicollinearity: The independent variables are not highly correlated with each other.
Violations of these assumptions can lead to biased or inefficient estimates and can affect the validity of inferences made from the model.
Read Chapter 11 in Statistical Analysis of Electronic Health Records,
pages 282 to 290
Coefficient of determination or R-squared is used to measure goodness
of fit between the model and the data. The statistic R^{2 }
measures the percentage of variation in the outcome (response variable in
the regression) explained by the independent variables. If a
regression has a low R-squared then the right variables have not been
included in the analysis, something often referred to as a model
specification bias.
Read Chapter 11 in Statistical Analysis of Electronic Health Records,
pages 277 to 280
Model Selection
To do regression, you have to try different mathematical models and see
which one fits the data best. A linear model is just a weighted sum
of independent variables. A non-linear model is a weighted sum of
independent variables and interaction terms. Interaction terms are
constructed as product of 2 or more independent variable.
Wikipedia: "All models are wrong is a common aphorism in statistics; it
is often expanded as "All models are wrong, but some are useful". The
aphorism is generally attributed to the statisticianGeorge
Box. The underlying concept, though, predates Box's writings."
Read Chapter 11 in Statistical Analysis of Electronic Health
Records page 274 to to 277
ChatGPT: In regression analysis, statistically significant variables
are those that have a strong and meaningful relationship with the
dependent variable, as determined by a hypothesis test. The coefficients
for these variables are estimated and their significance is determined by
the p-value. Variables with a low p-value (typically less than 0.05) are
considered statistically significant. In contrast, in a Markov blanket, a
variable is considered statistically significant if it is part of the set
of variables that shield a target variable from the effects of other
variables in a graph. The Markov blanket of a target variable is the set
of variables that, when known, renders the target variable independent of
all other variables. The variables in the Markov blanket are considered
statistically significant because they are the ones that are most directly
related to the target variable and can be used to make accurate
predictions about the target variable. In summary, in regression,
statistically significant variables are those that have a strong and
meaningful relationship with the dependent variable, as determined by a
hypothesis test; in a Markov blanket, a variable is considered
statistically significant if it is part of the set of variables that
shield a target variable from the effects of other variables in a graph.
Assignments should be submitted in Blackboard. The submission must
have a summary statement, with one statement per question. All
assignments should be done in Python, if possible. Include
at top of your report a summary section or include a file that reports the
summary of your work.
In the summary section, or summary page, write brief sentences comparing your work to answers given
or videos. For example, "I got the same answers as the Teach One
video for question 1."
Question 1: For the following X and Y data, calculate the regression equation. Plot the points and the line. Calculate the
residuals and sum of squared residuals. For solution see
required textbook pages 262 through 264.
Question 2: Using clean data in Question 1, change the
intercept of the estimated equation by +20% and by -20% and plot all
three lines. Calculate residuals for each of the three lines and
report which line has the lowest sum of squared residuals. Repeat
and this time change the coefficient in the equation by +20% and -20%.
Calculate the residuals for each of the three lines and report which
line has the lowest sum of squared residuals. See solution in page
264 exhibit 11.5 in the required textbook.
Question 3: Clean the Medical Foster Home
data. Limit the data to cost per day, patient disabilities in 365
days, survival, age of patients, gender of patients and
whether they participated in the medical foster home (MFH) program.
Clean the data using the following:
Remove all cases in which all values for disabilities in 365
days, age and gender, are missing. These are meaningless
data and should be dropped from analysis.
Remove any row in
which the treatment variable (MFH) is missing. MFH is an
intervention for nursing home patients. In this program, nursing home
patients are diverted to a community home and health care services are
delivered within the community home. The resident eats with the family
and relies on the family members for socialization, food and comfort.
It is called "foster" home because the family previously living in the
community home is supposed to act like the resident's family. Enrollment
in MFH is indicated by a variable MFH=1. A value of NaN or null is
missing value.
Various costs are reported in the file,
including cost inside and outside the organization.
Rely on cost per day. Exclude patients who have 0 cost per day within the
organization. These do not make sense. The cost is reported for specific time period after
admission, some stay a short time, and others some longer. Use daily cost so you do not
get caught on the issues related to lack of follow-up.
Select for your independent variables the probability of
disability in 365 days. These probabilities are predicted
from CCS variables. CCS in these data refer to Clinical
Classification System of Agency for Health Care Research and
Quality. CCS data indicate the comorbidities of the patient.
When null, it is assumed the patient did not have the comorbidity.
When data are entered it is assumed that the patient had the comorbidity
and the reported value is the first (maximum) or last (minimum) number of
days till admission to either the nursing home or the MFH. Thus an
entry of 20 under the minimum CCS indicates that from the most recent occurrence of the comorbidity
till admission was 20 days. An entry of 400 under the Maximum CCS indicates that from the
first time the comorbidity occurred till admission was 400 days. Because
of the relationship between disabilities and comorbidities, you
can rely exclusively on disabilities and ignore comorbidities.
Check if cases repeat
and should be deleted from the analysis.
In survival days, null
values indicate zero.
Convert all categorical variables to binary dummy variables.
For example, race has four values W, B, A, Other, and null value.
Create 5 binary dummy variables for these categories and use 4 of
them in the regression. For example, the binary variable
called Black is 1 when race is B, and 0 otherwise. In this
binary variable we are comparing all Black residents to non-Black
residents that include W, A, null, and other races.
In all variables where null value was not deleted row wise,
e.g. race being null, the null value should be made into a dummy
variable, zero when not null and 1 when null. Treat these
null variables as you would any other independent variable.
Gender is indicated as "M"
and "F"; revise by replacing M with 1 and F with 0.
Question 4: Visually check that cost is
normally distributed and see if log of cost is more normal than cost
itself. If a variable is not normally distributed, is the average of
the variable normal (see pages 261 in textbook)? Visually check that age and cost have a linear relationship.
Question 5: Regress cost per day on age (continuous
variable), gender (male=1, Female=0), survival, binary dummy variables for race, probabilities of functional disabilities,
and any null dummy variable you have created. Describe the
percent of variation explained, and F statistics. Show which variables have a
statistically significant effect on cost. Does age affect cost?
Does MFH reduce cost of care?
Question 6: In the following interactive graph, we
have fitted a regression line to data. Move one, and only one, point so that R2
declines to 0.05. Provide an image of the resulting data and line. Create
a concave curve with the data and report the fit between the regression
line and concave data. Attach a screen shot of your work.
Interactive Fit►
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For additional information (not part of the required reading), please see the following links: