Analysis

Formulate hypotheses using network terminology discussed throughout the course based on the following friendship relations.33 If you get stuck, it may be helpful to look at the terms included in the model below for hints on how to translate plain-language intuitions into network terminology. The first one is done for you:

1. Establishing and maintaining friendships takes time and resources. Students will not befriend people indiscriminately.

Hypothesis 1: Ties between students are not random.

Relational Hypotheses:

2. If a student nominates a person as a friend, that person is also likely to consider the student a friend.

Hypothesis 2: The friendship tie will be said to be reciprocal in this case.

3. Students will be friends with the friends of their friends (if A -> B and B -> C, then A -> C).

Hypothesis 3: This would point to a transitive friendship network.

4. Leaving school to find a place where smoking is permitted takes time that could otherwise be used for socializing. Students who smoke more will have less time to establish and maintain friendships.

Hypothesis 4:This would point to the degree centrality network of smokers suggesting that these smokers have a low rate of in-degree in terms of friendship.

5. Smoking is increasingly frowned upon in the U.S. Students who smoke more will likely not be very popular and few people will nominate them as friends.

Hypothesis 5:This would suggest a low rate of in degree centrailty for the smokers.

6. People with similar smoking behavior will be more likely to become friends.

Hypothesis 6:This can suggest how the non-smokers would create friendships with smokers and feel attracted to smoking in the later years of their life. Hence, it is the homophily that would exist because of the friendship network.

Smoking Behavior Hypotheses:

7. Students will likely smoke more as they get older.

Hypothesis 7: It is likely that the smokers who have not gone through any therapy or medication will increase their addiction toward smoking. This is to suggest that with time, smoking will increase.

8. Friendship relations will make students more similar in their smoking behavior.

Hypothesis 8:Based on the homphily network of friends, it is likely that actors will pick up the behavior habits of other actors in that network. This is to state that non-smokers will smoke like their smoker friends.

Analysis

Which two nodes represent isolates in the network at all three time periods? Nodes 13 and 20 are the isolates in all three networks over time. Describe the change in the network over time. Think about the formation of clusters and the incidence of smoking behavior. Time 1 graph depicts six clusters with actors V 19, V 11 and V 41 acting as the bridge in the network. This is to further explain the actors: V 41, V 44, V 15, V 16, V 1 as light smokers, actors: V 11, V 2, V 26, V 42, V 12, V 23, and V 8 as heavy smokers and, the rest of the actors in the network as non-smokers. It is interesting to note the behavior of actors V 26 and V 29 from Time 1 to Time 3. V 19 is one of the bridge actors in Time 1, while in Time 2 and 3, this actor is a heavy smoker. This suggests that the homophily and addiction along with increase of their behavior has increased over time in this actor’s network. Further actor 8 is noted as a heavy smoker throughout in all three networks. It was an isolate in Time 1, became friends with a light smoker V 6 affecting that actor to an extent that V 6 can be noted a heavy smoker in Time 3 network. It is fascinating to note that actor V 24 who is a non-smoker is friends with a heavy smoker V 23. In all three networks, V 24 remains unaffected with intercation with either V 23 (heavy smoker), V 27, a light smoker in Time 2, or even V 6, a heavy smoker in Time 3 network.

Using the three visualizations, evaluate hypothesis five. The actor V 23 can be used as an example to understand hypthesis 5 that smokers will have a low in-degree centrailty of friendship in their network. Actor 23 remains friends with heavy smokers in Time 2 and 3. # Creating the SIENA Model

To build a SIENA model, we need to create dependent variables, explanatory variables, a combination of both types of variables, and our model specification.

First, we create a SIENA data object including the longitudinal friendship network and the smoking behavioral variable. The results of creating that model, smokeBehXfriendship:

## Dependent variables:  friendship, smokingbeh 
## Number of observations: 3 
## 
## Nodeset                  Actors 
## Number of nodes              50 
## 
## Dependent variable friendship      
## Type               oneMode         
## Observations       3               
## Nodeset            Actors          
## Densities          0.046 0.047 0.05
## 
## Dependent variable smokingbeh
## Type               behavior  
## Observations       3         
## Nodeset            Actors    
## Range              1 - 3

Using our hypotheses above, we will construct a list of parameters to test using our Siena model. A table of those parameters follows:

##    name       effectName                          include fix   test 
## 1  friendship constant friendship rate (period 1) TRUE    FALSE FALSE
## 2  friendship constant friendship rate (period 2) TRUE    FALSE FALSE
## 3  friendship outdegree (density)                 TRUE    FALSE FALSE
## 4  friendship reciprocity                         TRUE    FALSE FALSE
## 5  friendship smokingbeh alter                    TRUE    FALSE FALSE
## 6  friendship smokingbeh ego                      TRUE    FALSE FALSE
## 7  friendship same smokingbeh                     TRUE    FALSE FALSE
## 8  smokingbeh rate smokingbeh (period 1)          TRUE    FALSE FALSE
## 9  smokingbeh rate smokingbeh (period 2)          TRUE    FALSE FALSE
## 10 smokingbeh smokingbeh linear shape             TRUE    FALSE FALSE
## 11 smokingbeh smokingbeh quadratic shape          TRUE    FALSE FALSE
## 12 smokingbeh smokingbeh total similarity         TRUE    FALSE FALSE
##    initialValue parm
## 1     4.69604   0   
## 2     4.32885   0   
## 3    -1.46770   0   
## 4     0.00000   0   
## 5     0.00000   0   
## 6     0.00000   0   
## 7     0.00000   0   
## 8     0.81720   0   
## 9     0.43579   0   
## 10   -0.22314   0   
## 11    0.00000   0   
## 12    0.00000   0

Next, we will create our model. You can learn about the function that creates Siena models by typing ?sienaModelCreate into your R console.

## Estimates, standard errors and convergence t-ratios
## 
##                                                Estimate   Standard   Convergence 
##                                                             Error      t-ratio   
## Network Dynamics 
##    1. rate constant friendship rate (period 1)   5.1870 ( NA       )   -0.1013   
##    2. rate constant friendship rate (period 2)   4.0660 ( NA       )   -0.4282   
##    3. eval outdegree (density)                  -2.9894 ( NA       )    0.2552   
##    4. eval reciprocity                           2.7120 ( NA       )   -0.4172   
##    5. eval smokingbeh alter                     -0.3817 ( NA       )   -3.7410   
##    6. eval smokingbeh ego                        0.4047 ( NA       )   -0.9260   
##    7. eval same smokingbeh                       0.8677 ( NA       )    1.0466   
## 
## Behavior Dynamics
##    8. rate rate smokingbeh (period 1)          129.4845 ( NA       )             
##    9. rate rate smokingbeh (period 2)          137.7225 ( NA       )             
##   10. eval smokingbeh linear shape             -32.5199 ( NA       )   -5.8516   
##   11. eval smokingbeh quadratic shape           35.1530 ( NA       )   -2.7810   
##   12. eval smokingbeh total similarity           0.2780 ( NA       )    1.3680   
## 
## Overall maximum convergence ratio:        NA 
## 
## 
## Warning: *** Warning: Noninvertible estimated covariance matrix *** 
## 
## Total of 3060 iteration steps.

Analysis

Has your model converged sufficiently? If not, note which terms have not converged, rerun your model using the previous Siena model values as your starting point, and re-evaluate. Repeat this process until the overall maximum convergence ratio and the convergence t-ratio for each term are within acceptable levels.44 If your model has not converged, uncomment prevAns = ans1 in the code block titled Model Creation and rerun that block of code and print results. This will use the previous values generated by the previous model creation as the starting point in the estimation and proceed through the model construction process again. See pp. 58—59 of the RSiena manual for more information.

The Model has converged sufficiently according to the requirements above. The individual t-ratios are less than the absolute value of 0.1.This is to explain that these individual t-ratios are: -0.0008, 0.0240 ,0.0437, 0.0509, -0.-417, -0.0589, 0.0428, -0.0121, -0.0216, -0.0060, -0.0480, and 0.0756, which are definitely less than the absolute value of 0.1. Further, since the maximun convergence ratio is 0.1630, which is less than 0.25, the model has converged sufficiently.
# Understanding the Estimate Column

The Estimate column, also reported as the theta vector within an RSiena object’s effects vector, represents the chance of an actor forming a tie within the network based on interactions within and between networks or in relation to the presence or absence of a behavior. The Standard Error column provides information about the amount of variation among actors within the network on the given parameter.

Analysis

For each of your hypotheses, indicate which parameter operationalizes your hypothesis. Using the tables created above, evaluate that hypothesis, and report whether your results were significant.55 If you’re having difficulty matching up parameters with your hypotheses, take a look at pp. 41-49, § 6.2, in the RSiena Manual.

Hypothesis 1: This is stated above as: ties between students will not tend to be random suggesting that friendship networks will less likely change over time. The Constant Friendship rate for Period 1 and Period 2, both are more than 2 and positive. Since the values are significant, this suggests that the parameter operationalizes hypothesis 1. As an example, when the absolute value for Constant Friendship rate for Period 1: 5.7749 is divided by Standard Error: 0.8901, 5.7749/0.89012 > 2.

Hypothesis 2: This is stated as: friendship ties will tend to be reciprocal suggesting that the significant value for reciprocity is positive (2.7692) and standard error is 0.2093 greater than 2. This suggests that the parameter operationalizes hypothesis 2.

Hypothesis 3: N/A

Hypothesis 4: This is stated as: smokers will tend to experience a low degree centrality in their network in comparison to the non-smoker actors. This suggests that since the value for Smoking Behavior Ego is < 2 (0.1161 divided by 0.1846) hence, the parameter does not operationalize hypothesis. This is to further state that as seen in the network for Time 1, Time 2 and Time 3, respetively, heavy smokers are friends with a number of other actors and are not isolates.

Hypothesis 5: N/A

Hypothesis 6: This is stated as: Homphily tends to give rise to more friendship ties in a network. This suggests that since the absolute value for Smoking Behavior total similarity is 1.1347 which when divided by the Standard Error 0.6272 is less than 2, the parameter does not operationalize hypothesis 6.

Hypothesis 7: This is stated as: the behavior in smokers will increase over time. This suggests that the value for Constant Friendship rate for Period 1 > 2 but the value for Constant Friendship rate for Period 2< 2, which does not support the hypothesis. Hence, the value for Smoking Behavior Quadratic Shape (Absolute value) 1.9128 divided by (Standard Error) 0.4307 is greater than 2, the value is said to be positive as well as > 2. Hnece, the parameter does operationalizes for hypothesis 7.

Hypothesis 8: This is stated as: non-smokers will tend to become light smokers and the light smokers will tend to become heavy smokers. This further suggests that since the smoking behavior alter remains 0.1118 with error 0.1627, which is below 2, stating that this behavior will not be the dominating behavior among all actors in the networks, hence the parameter does not operationalizes hypothesis 8.