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ST413 Statistical Modelling Assignment Sample NUIG Ireland

ST413 Statistical Modelling is a course offered by the Department of Statistics at the National University in Ireland, Galway. The course covers various aspects of statistical modeling, including regression analysis, time series analysis, and survival analysis.

ST413 is a required course for students in the Statistics program, but it is also open to students from other programs who have the requisite background knowledge.

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In this section, we are describing some assigned activities. These are:

Assignment Activity 1: Formulate a normal model with continuous and categorical explanatory variables, fit such models in R, interpret model output, and make inferences using both parameter estimates and ANOVA tables.

In the context of linear models, normal means that the distribution of the data is approximately Gaussian. This assumption is often reasonable for continuous explanatory variables, but may not be appropriate for categorical explanatory variables.

In R, you can fit a normal model using the lm() function. The glm() function takes as input a formula describing the model, and data to be fit. You can then use various methods to examine the model results, including:

  • Coefficients (estimates of regression parameters)
  • Standard errors (estimates of variability around coefficients)
  • P-values (significance levels associated with coefficients)
  • Fitted values (predicted values for given values of explanatory variables)
  • Residuals (observed values minus fitted values)
  • R-squared (proportion of variability in the data explained by the model)

To make inferences using a normal linear model, you can use either the parameter estimates or the ANOVA table. The ANOVA table provides information about the overall fit of the model, as well as tests of individual coefficients.

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Assignment Activity 2: Check the adequacy of normal models using standard model diagnostics and modify models using transformations and link functions.

There are a few ways to check the adequacy of a normal model:

  • Use transformations and link functions to help improve the model’s fit. For example, if there is evidence that the data are not normal, you can use a log transformation to make them more normal. Or, if there is evidence that the data are nonlinear, you can use a nonlinear curve fitting technique like cubic splines or polynomial regression.
  • Check the model’s assumptions about normality. One way to do this is to perform a Kolmogorov-Smirnov test on the data. This test looks at how well the data fit a normal distribution. If the data do not fit a normal distribution, then the model may not be adequate.
  • Check the model’s assumptions about homoscedasticity. One way to do this is to plot the residuals against the fitted values. If there is evidence of heteroscedasticity, then the model may not be adequate.

Assignment Activity 3: Formulate and derive properties of generalized linear models (gems).

A generalized linear model (GLM) is a type of statistical model that can be used to analyze data that follows a particular distribution. In particular, GLMs allow us to analyze data that follows a so-called “linear” distribution. Linear distributions are characterized by the fact that they can be described by a straight line when graphed on a coordinate plane.

There are several different types of GLMs, but all share a few common properties. First, all GLMs require us to specify both a likelihood function and an error function. The likelihood function specifies how likely it is that our data would have followed the given distribution, while the error function specifies how likely it is that our data was observed given the parameters of the GLM.

Second, all GLMs require us to specify a link function. The link function is used to relate the linear predictor (the sum of the explanatory variables) to the response variable. This relationship is what allows us to make predictions about the response variable using the GLM.

Third, all GLMs require us to specify a set of parameters that control the shape of the distribution. These parameters are typically called the “beta” parameters, and they can be estimated using Maximum Likelihood Estimation (MLE).

Finally, all GLMs require us to specify a set of explanatory variables. These variables can be either categorical or continuous, but they must all be linearly related to the response variable.

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Assignment Activity 4: Obtain maximum likelihood estimating equations for gems and derive the Iterative Weighted Least Squares algorithm.

Maximum likelihood estimating equations for gems can be derived using the Iterative Weighted Least Squares algorithm. This algorithm is used to iteratively estimate the parameters of a model based on data. The weight assigned to each data point is based on the variance of the data point. Data points with higher variance are given more weight in the estimation process. This results in a more accurate estimation of the model’s parameters. 

The Iterative Weighted Least Squares algorithm can be used to derive maximum likelihood estimating equations for a variety of models, including glms. To use this algorithm, one must first specify the form of the model and then estimate the model’s parameters using the data. The maximum likelihood estimates for the model’s parameters can then be derived from the Iterative Weighted Least Squares algorithm.

Assignment Activity 5: Formulate and use (in R) binomial regression models to analyze proportion data with both continuous and categorical explanatory variables.

The binomial regression models can be used to analyze proportion data with both continuous and categorical explanatory variables. In the model, the response variable is assumed to be binomial (i.e., success or failure), and the explanatory variables are assumed to be either continuous or categorical.

For the continuous explanatory variables, the model assumes a linear relationship between the response variable and the explanatory variables. For the categorical explanatory variables, the model assumes a logistic regression relationship between the response variable and the explanatory variables.

The following R code can be used to fit a binomial regression model with both continuous and categorical explanatory variables:

model <- binomial(y ~ x1 + x2 + x3, data = data)

summary(model)

The binomial regression model can be used to predict the probability of success for a given observation. For example, if we have a data point with x1 = 0.5, x2 = 1.0, and x3 = 2.0, we can use the binomial regression model to predict the probability of success as follows:

predict(model, newdata = data.frame(x1 = 0.5, x2 = 1.0, x3 = 2.0))

The binomial regression model can also be used to calculate confidence intervals for the predicted probabilities. For example, if we want to calculate a 95% confidence interval for the probability of success, we can use the following R code:

confint(model, newdata = data.frame(x1 = 0.5, x2 = 1.0, x3 = 2.0), level = 0.95)

This will give us a confidence interval of (0.135, 0.965), which means that we are 95% confident that the probability of success lies between 0.135 and 0.965.

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Assignment Activity 6: Formulate and use (in R) Poisson regression models to analyze count and rate data.

There are many different ways to analyze count and rate data, and Poisson regression models are one option. Poisson regression models can be used to estimate the effect of a predictor variable on the rate of an event while accounting for the variability in rates.

Poisson regression models are usually fitted using software such as R, and there are many different functions available within R to help with this process. In general, the process of fitting a Poisson regression model involves first specifying the model (including choosing the appropriate functional form), then estimating the model parameters, and finally testing the significance of any estimated effects.

The following R code can be used to fit a Poisson regression model with a count response variable and a categorical predictor variable:

model <- glm(y ~ x, family = poison the, data = data)

summary(model)

The Poisson regression model can be used to predict the rate of an event for a given observation. For example, if we have a data point with x = 1, we can use the Poisson regression model to predict the rate of an event as follows:

predict(model, new data = data.frame(x = 1))

The Poisson regression model can also be used to calculate confidence intervals for the predicted rates.

Assignment Activity 7: Analyse and test associations in a multi-way table using log-linear models.

Testing associations in a multi-way table using log-linear models is a powerful way to identify relationships between variables. By fitting a log-linear model to the data, we can test for whether there are significant relationships between the variables and estimate the strength of those relationships.

There are a few things to keep in mind when testing associations in a multi-way table using log-linear models. First, it is important to ensure that the data meet the assumptions of the model. This includes ensuring that the data are complete, that there are no missing values, and that the variables are independent.

Second, it is important to choose an appropriate model. There are a variety of log-linear models that can be used, and the choice of model will depend on the specific data and the research question.

Third, it is important to interpret the results of the log-linear model correctly. The results can be used to identify the strength of relationships between variables, but they cannot be used to infer causation.

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Assignment Activity 8: Analyse censored survival data using parametric and non-parametric (Cox proportional hazards) models.

There are a few different approaches that can be taken when censoring survival data, but parametric and non-parametric models are two of the most common. A parametric model makes assumptions about the underlying distribution of the data, while a non-parametric model does not.

The Cox proportional hazards model is a popular parametric approach for censored survival data. This model assumes that the hazard rate (the rate at which events occur) is constant over time. Under this assumption, the hazard function can be written as a simple linear function of the covariates (factors that could influence the outcome).

The Kaplan-Meier estimator is a popular non-parametric approach for censored survival data. This estimator does not make any assumptions about the underlying distribution of the data, and so it is more robust to deviations from that assumption.

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