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Achievement StandardAS 91581 PPDAC cycleOne of the principles:Grade distinctions should not be based on the candidate being required to acquire and retain more subject-specific knowledge. ASAME3.9Investigate bivariate measurement dataInvestigate bivariate measurement data, with justificationInvestigate bivariate measurement data, with statistical insight3Assessment is based on the PPDACPose a questionPlanDataAnalyseConclude

Back to questionStatistical enquiry cycle (PPDAC)

5The Statistical Enquiry Cycle (PPDAC)PPDAC is vital

6Using the statistical enquiry cycle to investigate bivariate measurement data involves:posing an appropriate relationship question using a given multivariate data setselecting and using appropriate displaysidentifying features in datafinding an appropriate modeldescribing the nature and strength of the relationship and relating this to the contextusing the model to make a predictioncommunicating findings in a conclusion7From Explanatory Note 3

Using the statistical enquiry cycle to investigate bivariate measurement data involves:posing an appropriate relationship question using a given multivariate data setselecting and using appropriate displaysidentifying features in datafinding an appropriate modeldescribing the nature and strength of the relationship and relating this to the contextusing the model to make a predictioncommunicating findings in a conclusion

8Posing relationship questions Possibly the most important component of the investigation Time spent on this component can determine the overall quality of the investigationThis component provides an opportunity to show justification (M) and statistical insight (E)

9Posing relationship questions What makes a good relationship question?It is written as a question.It is written as a relationship question.It can be answered with the data available.The variables of interest are specified.It is a question whose answer is useful or interesting.The question is related to the purpose of the task.Think about the population of interest. Can the results be extended to a wider population?

10Data

WHOA! We need to understand the variables before we can make any progress.

Consider the variables (using context))

Think about which variables could be related JUSTIFY YOUR REASONING

Developing question posing skills Pose several relationship questions (written with reasons/justifications)

Possibly critique the questions

The precise meaning of some variables may need to be researched16Different relationship questions Is there a relationship between variable 1 and variable 2 for Hectors dolphins?

What is the nature of the relationship between variable 1 and variable 2 for Hectors dolphins?

Can variable 1 be used to predict variable 2 for Hectors dolphins?

17Research about the situation helps develop ideasWe might like to ask ourselves why we would want to know about any relationships that might existReference to research19

HINT: GroupingsA morphological study of skull and mandible features was undertaken to examine variation between the most genetically distinct population, occurring on the west coast of the North Island, and the populations around the South Island. Univariate and principal component analyses demonstrate that the North Island population can be differentiated from the southern populations on the basis of several skeletal characters.Developing question posing skills It is expected that you do some research:

Improve knowledge of variables and context

May find some related studies that creates potential for integration of statistical and contextual knowledge22

Developing question posing skills Draw some scatter plots to start to investigate your questions

Reduce, add to and/or prioritise their list of questions

Possibly critique the questions again23Consider the variables (using context))

Appropriate displays Which variable goes on the x-axis and which goes on the y-axis?It depends on the question and on the variables of interest

Is there a relationship between zygomatic width and rostrum length for Hectors dolphins?

25Variables on axes Is there a relationship between zygomatic width and rostrum length for Hectors dolphins?

26This is a straight forward relationship question.Is there a relationship between zygomatic width and rostrum length for Hectors dolphins?

27This is a correlation type question.Is there a relationship between zygomatic width and rostrum length for Hectors dolphins?

28Is there a relationship between zygomatic width and rostrum length for Hectors dolphins?There is no difference in the roles of each variable so it does not matter which variable goes on the x-axis and which goes on the y-axis.

For this question it does not matter which variable goes on each axis.

Which variable goes on the x-axis and which goes on the y-axis?It depends on the question and on the variables of interestIs there a relationship between zygomatic width and rostrum length for Hectors dolphins?

Is there a relationship between rostrum width at midlength and rostrum width at base for Hectors dolphins?

30Variables on axes Is there a relationship between rostrum width at midlength and rostrum width at the base for Hectors dolphins?

31I think that the width at the base could help determine (or influence) the width at midlength.Is there a relationship between rostrum width at midlength and rostrum width at the base for Hectors dolphins?

32Rostrum length at base (RWB) the explanatory variable and rostrum width at midlength the response variable..Is there a relationship between rostrum width at midlength and rostrum width at the base for Hectors dolphins?

33Which variable goes on the x-axis and which goes on the y-axis?It depends on the question and on the variables of interestIs there a relationship between zygomatic width and rostrum length for Hectors dolphins?Is there a relationship between rostrum width at midlength and rostrum width at the base for Hectors dolphins?For Hectors dolphins, can rostrum length be used to predict mandible length?

34Variables on axes For Hectors dolphins, can rostrum length be used to predict mandible length?

35This is a regression questionFor Hectors dolphins, can rostrum length be used to predict mandible length?

36It is clear that we are interested in how mandible length responds to rostrum length, so rostrum length is the explanatory variable and mandible length is the response variable.

37Experiments: If the data comes from an experiment then some variables would be classed as input variables and others as output variables. It would make sense to see how an input variable affects an output variable. The input variable would be the explanatory variable and the output variable would be the response variable.

Using the statistical enquiry cycle to investigate bivariate measurement data involves:posing an appropriate relationship question using a given multivariate data setselecting and using appropriate displaysidentifying features in datafinding an appropriate modeldescribing the nature and strength of the relationship and relating this to the contextusing the model to make a predictioncommunicating findings in a conclusion

39Same approach whether linear or non-linearinvestigate bivariate measurement data involves:posing an appropriate relationship question using a given multivariate data setselecting and using appropriate displaysidentifying features in datafinding an appropriate modeldescribing the nature and strength of the relationship and relating this to the contextusing the model to make a predictioncommunicating findings in a conclusion

40Features, model, nature and strength Generate the scatter plot using iNZight(or excel)RWM vs RWB

41Use iNZightDo RWM versus RWB in iNZight

Features, model, nature and strength Generate the scatter plotLet the data speakUse your eyes (visual aspects)

Write about what you see, not what you dont seeDONT fit a model yet

42Features, model, nature and strength Template for features (but allow flexibility)TrendAssociation (nature)Strength (degree of scatter)Groupings/clustersUnusual observationsOther (e.g., variation in scatter)If there are groupings evident in the plot then it may be better to explore the groups separately at this stage.

43Trend From the scatter plot it appears that there is a linear trend between rostrum width at base and rostrum width at midlength.Use descriptions of variables rather than variable names- keeps it contextual.This is a reasonable expectation because two different measures on the same body part of an animal could be in proportion to each other.

44Key point is linear or non-linear.Use of descriptions of variables rather than variable names.Refer to the graph.

Last paragraph: This illustrates some reflection.

Association The scatter plot also shows that as the rostrum width at base increases the rostrum width at midlength tends to increase.This is to be expected because dolphins with small rostrums would tend to have small values for rostrums widths at base and midlength and dolphins with large rostrums would tend to have large values for rostrums widths at base and midlength.

45A contextual description is preferable to one using technical terms.However it is appropriate to use terms such as positive, negative or no association, but they are better used after the contextual description.Tends to is a good term to use.Higher level considerationsYou should reflect on the nature of the relationship with respect to the context.At this stage you could acknowledge (if the data does not come from a randomised experiment) that they have found only a statistical relationship and that this does not necessarily imply a causal relationship between the variables.Alternatively, if the data comes from a suitable experiment they could make a causation claim in their conclusion.Students may acknowledge that other variables (which they must name) would impact on the (response) variable, and suggest how they might impact on the variable. For example, gender, age, etc.

Association The scatter plot also shows that as the rostrum width at base increases the rostrum width at midlength tends to increase.A contextual description is preferable to one using technical terms.However it is appropriate to use terms such as positive, negative or no association, but they are better used after the contextual description.

46Higher level considerations This is to be expected because dolphins with small rostrums would tend to have small values for rostrums widths at base and midlength and dolphins with large rostrums would tend to have large values for rostrums widths at base and midlength.You should reflect on the nature of the relationship with respect to the context.At this stage you could acknowledge (if the data does not come from a randomised experiment) that they have found only a statistical relationship and that this does not necessarily imply a causal relationship between the variables.

47Higher level considerations Alternatively, if the data comes from a suitable experiment you could make a causation claim in your conclusion.You may acknowledge that other variables (which they must name) would impact on the (response) variable, and suggest how they might impact on the variable. For example, gender, age, etc. and perhaps show these and compare.

48Find a model Because the trend is linear I will fit a linear model to the data.You must state why you have selected this particular model. The line is a good model for the data because for all values of rostrum width at base, the number of points above the line are about the same as the number below it.

49Find a model Because the trend is linear I will fit a linear model to the data.You must state why you have selected this particular model. Dont show the equation yet. There are still features in the data to comment on.

50Find a model Because the trend is linear I will fit a linear model to the data.You must state why you have selected this particular model.

A discussion of fit throughout the range of x-values is required.If the number of observations is small that casts some doubt on the reliability of the model.

51Find a model The line is a good model for the data because for all values of rostrum width at base, the number of points above the line are about the same as the number below it.

A discussion of fit throughout the range of x-values is required.

If the number of observations is small that casts some doubt on the reliability of the model.52Strength The points on the graph are reasonably close to the fitted line so the relationship between rostrum width at midlength and rostrum width at base is reasonably strong. This is supported by the correlation coefficient r = This must refer to visual aspects of the display.Appropriate descriptors are: strong, moderate or weak.

53Strength The points on the graph are reasonably close to the fitted line so the relationship between rostrum width at midlength and rostrum width at base is reasonably strong. This is supported by the correlation coefficient r = You must refer to the degree of scatter about the trend or, equivalently, the closeness of the points to the trend.

54Groupings

None are apparent from the scatter plot.55If groupings are apparent then can a reason be found that explains the groupings?

None are apparent from the scatter plot.56But the data set has an obvious grouping variable (Island).

None are apparent from the scatter plot.57INZightGroupings

59Groupings Now it seems that NI Hectors dolphins seem to have larger rostrum widths at base than SI ones.This is now helpful for doing predictions. There are some common values of RWB for both species of dolphin.This allows more comments to be made about the model fitted to all data points.

60Groupings Now it seems that NI Hectors dolphins seem to have larger rostrum widths at base than SI ones.Many NI points are above the line.Why is that?It could be the effect on the fitted line of the unusual point (RWB = 86, RWM = 50).

61Use iNZight to look at NI and SI dolphins separately (Subset by Island)

Groupings

62This now provides an opportunity to comment on these models.

63South Island dolphins: Model seems appropriate.

64North Island: Only 13 data points model may not be useful for reliable conclusions.

65Points with lower RWB values tend to be below the line, points in the middle tend to be above.Could try a quadratic model but small number of data points is an issue

66Unusual points One dolphin, one of those with a rostrum width at base of 86mm, had a smaller rostrum width at midlength compared to dolphins with the same, or similar,rostrum widths at base.

67Refer to actual data points.If not discussed earlier, comment on the effect any unusual values would have on the model.

Unusual points One dolphin, one of those with a rostrum width at base of 86mm, had a smaller rostrum width at midlength compared to dolphins with the same, or similar,rostrum widths at base. Refer to actual data points.

68Unusual points One dolphin, one of those with a rostrum width at base of 86mm, had a smaller rostrum width at midlength compared to dolphins with the same, or similar,rostrum widths at base. Comment on the effect any unusual values would have on the model.

69Anything else Variation in scatter?

70Anything else Variation in scatter?

Constant versus non-constant scatter relates to assumptions of the linear regression i.e. that the residuals are normally distributed.

71Anything else Variation in scatter?

Variation in scatter is relevant when discussing precision of predictions.

72Variation in scatter is more cognitively demanding because it involves distribution reasoning.Dont be distracted by scarcity of data.This must refer to visual aspects of the display.Students should try to keep this contextual. For example: As the values of x increase the amount (or degree) of variation in y tends to increase (for fanning out)

Anything else Variation in scatter?

Refer to visual aspects of the display and keep it contextual.For example: As the values of x increase the amount (or degree) of variation in y tends to increase (for fanning out)

73Dont be distracted by scarcity of data.

Prediction

74Try to use relevant values of x. In this case any RWB value from 81mm to 86mm is sensible.

75Prediction Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

76Use all three models with RWB = 85mm (all, NI only, SI only) Using RWB = 85mmAll points: RWM = 0.77 x 85 8.72 = 56.73NI dolphins: RWM = 0.48 x 85 + 19.19 = 59.99SI dolphins: RWM = 0.46 x 85 + 14.37 = 53.47

Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

77Interpret in context, with units.Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

78Use sensible rounding.Using RWB = 85mmAll points: RWM = 0.77 x 85 8.72 = 56.73NI dolphins: RWM = 0.48 x 85 + 19.19 = 59.99SI dolphins: RWM = 0.46 x 85 + 14.37 = 53.47

Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

79Dangers predicting outside the range of observed x-values.Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

80How good is a prediction?Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

A prediction is an estimate so I need to consider bias and precision.81How good is a prediction?Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

Bias:If a model is good, then a prediction is likely to be accurate82How good is a prediction?Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

The number of data points used to form the model is relevant.83How good is a prediction?Linear TrendRWM = 0.77 * RWB + -8.72

Summary for Island = 1 Linear TrendRWM = 0.48 * RWB + 19.19Summary for Island = 2 Linear TrendRWM = 0.46 * RWB + 14.37

Precision Relates to the degree of scatterDont relate a prediction to observed y-values.84Statistical enquiry cycle (PPDAC)

85Communicating findings in a conclusion Each component of the cycle must be communicatedThe question(s) must be answered86Summary Basic principlesEach componentContextVisual aspectsHigher level considerationsJustifyExtendReflect87Other issues (if time) The use or articles or reports to assist contextual understandingHow to develop understanding of outliers on a modelThe place of residuals and residual plotsIs there a place for transforming variables?88